Chapter 0: Preliminaries

• Section A: Overview of the course
  • Symmetries of objects
  • Finding roots of polynomials
  • Uniform models to avoid repetition

• Section B: Useful tools on sets and functions
  • Properties of sets
    • Thm 0.1: Distributive laws: A ∩ (∪_α B_α) = ∪_α(A ∩ B_α) and A ∪ (∩_α B_α) = ∩_α(A ∪ B_α)
    • Thm 0.2: De Morgan's laws: A - (∪_α B_α) = ∩_α(A - B_α) and A - (∩_α B_α) = ∪_α(A - B_α)
    • Template to prove two sets are equal
    • Def 0.3: The Cartesian product A × B := {(a,b) | a ∈ A, b ∈ B} and ∏_{α ∈ J} A_α := {(a_α)_{α ∈ J} | a_α ∈ A_α}
    • Lemma 0.4: If A ⊆ G and B ⊆ H, then A × B ⊆ G × H.
    • Axiom of Choice: If A_α ≠ ∅ for all α ∈ J, then ∏_{α ∈ J} A_α ≠ ∅.
    • Def 0.5: The power set P(S) of a set S is the set of all subsets of S.
    • Discussion of notation
  • Properties of functions
    • Def 0.6: For a function f: G → H, the set G is the domain, the set H is the codomain.
      The image of a subset A of G is f(A) := {f(a) | a ∈ A}.
      The preimage of a subset B of H is f ^-1(B) := {g | g ∈ G and f(g) ∈ B}.
      The image of f is f(G).
    • Thm 0.7: (PAN = ``Preimages are nice''): If f: X → Y, E ⊆ E' ⊆ Y, and E<sub>α</sub> ⊆ Y for all α, then (a) f <sup>-1</sup>(E) ⊆ f <sup>-1</sup>(E'), (b) f <sup>-1</sup>(∪<sub>α</sub> E<sub>α</sub>) = ∪<sub>α</sub>f <sup>-1</sup>(E<sub>α</sub>), and (c) f <sup>-1</sup>(∩<sub>α</sub> E<sub>α</sub>) = ∩<sub>α</sub>f <sup>-1</sup>(E<sub>α</sub>).
    • Thm 0.8: (``Images are sometimes nice''): If f: X → Y, C ⊆ C' ⊆ X, and C_α ⊆ X for all _α, then (a) f(C) ⊆ f(C'), (b) f(∪_α C_α) = ∪_α f(C_α), and (c) f(∩_α C_α) ⊆ ∩_α f(C_α).
    • Thm 0.9: (a) A ⊆ f ^-1(f(A)), with = if f is injective. (b) f(f ^-1(B)) ⊆ B, with = if f is surjective.
    • Template to prove two functions are equal
    • Examples
    • Def 0.10: A function f: G → H is well-defined if whenever g,g' ∈ G and g = g', then f(g) = f(g').
    • Rmk: All functions are well-defined.
    • Def 0.11: Let f: X → Y be a function.
      The function f is one-to-one (also called an injection) if whenever x,x' ∈ X and f(x) = f(x') then x = x'.
      The function f is onto (also called a surjection) if for every y in Y, there is an x in X with f(x) = y.
      The function f is a bijection if f is both one-to-one and onto.
      The function f is invertible if there is a function g:Y → X such that f ∘ g = 1_Y and g ∘ f = 1_X.
    • Thm 0.12: Let a: G → H, b: H → K, and c: K → L be functions. Then:
      (1) If a and b are one-to-one then ba is one-to-one.
      (1a) If ba is one-to-one then a is one-to-one.
      (2) If a and b are onto then ba is onto.
      (2a) If ba is onto then b is onto.
      (3) a is a bijection if and only if a is invertible.
    • Examples
  • Equivalence relations
    • Def 0.13: An equivalence relation ∼ on a set G is a subset R of G × G (where (a,b) ∈ R is written a ∼ b) that satisfies the following for all a,b,c in G:
      (1) a ∼ a (reflexive),
      (2) a ∼ b implies b ∼ a (symmetric), and
      (3) a ∼ b and b ∼ c implies a ∼ c (transitive).
      The equivalence class of an element a of G is [a] := {b | b ∼ a}. The notation G/∼ denotes the set of equivalence classes.
    • Def 0.14: A partition of a set G is a collection of nonempty disjoint subsets of G whose union is G.
    • Examples
    • Prop 0.15: Let G be a set.
      (1) If ∼ is an equivalence relation on G, then G/∼ is a partition of G.
      (2) If S is a partition of G, and T = {(g,h) ∈ G × G | g,h are in the same element of S}, then T is an equivalence relation on G.
    • Cor 0.16: Let f: G → H be a function, and let R = {(g,g') ∈ G × G | f(g) = f(g')}. Then R is an equivalence relation on G.
    • Thm 0.17: (Function Building Theorem (FBT) for sets): Let ∼ be an equivalence relation on a set X, and let f: X → Y be a function satisfying the property that whenever x,x' ∈ X and x ∼ x' then f(x) = f(x'). Then:
      (1) There is a well-defined function g:X/∼ → Y defined by g([x]) = f(x) for all [x] in X/∼.
      (2) If f is onto, then g is onto.
      (3) If f also satisfies the property that whenever x,x' ∈ X and f(x) = f(x') then x ∼ x', then g is one-to-one.
    • Examples
    • Template for proving a well-defined function with domain X/∼.
  • Cardinality
    • Def 0.18: A set G is finite if there is a bijection G → {1,...,n} for some natural number n, or G is empty. In this case the number n is called the cardinality of G.
      A set G is infinite if G is not finite.
    • Thm 0.19: Let G be a set.
      (1) The following are equivalent: (1a) G is finite. (1b) There is an onto function {1,...,n} ↠ G for some n ∈ N. (1c) There is a one-to-one function G ↪ {1,...,n} for some n ∈ N.
      (2) The following are equivalent: (2a) G is infinite. (2b) There is an onto function G ↠ N. (2c) There is a one-to-one function N ↪ G.
    • Thm 0.20: Let G and H be finite sets and let G' be a subset of G. Then G', G ∩ H, G ∪ H, and G × H are finite sets.
    • Examples

Chapter 1: Introduction to groups

• Section A: Definitions and first examples
  • Motivation: Studying symmetries of objects
  • Def 1.1: A binary operation on a set G is a function G × G → G.
  • Def 1.2: A group is a set G with a binary operation (called group multiplication; the operation maps an ordered pair (a,b) to an element of G denoted ab) satisfying the following:
    (1) For all a,b,c in G, (ab)c = a(bc). (associative)
    (2) There is an element e such that ae = a = ea for all a in G.
    (3) For each element a in G, there is an element b in G such that ab = e = ba.
  • Examples:
    • Z,Q,R,C with + and Z/nZ with + modulo n
    • Z^x={1,-1}, Q^x=Q-{0}, R^x=R-{0}, C^x=C-{0} with ⋅ and U(n)=(Z/nZ)^x= {k ∈ {1,...,n-1} | gcd(k,n)=1} with ⋅ modulo n
    • Def 1.4: Let X be a set. Perm(X) = {f:X → X | f is a bijection} with ∘ (composition) is the permutation group of X.
      • Def 1.5: For a subset X of Euclidean space Rⁿ, a symmetry X is a bijection f: X → X that preserves Euclidean distance d (that is, d(p,q)=d(f(p),f(q)) for all p,q ∈ R²), and the group of symmetries of X is the set {f:X → X | f is a bijection and f preserves Euclidean distance} with group operation of function composition.
    • Def 1.6: The dihedral group of order 2n, D_2n, is the group of symmetries of the regular n-gon in R².
    • Def 1.7: For R ∈ {Z,Q,R,C,Z/nZ}, the general linear group GL_n(R) is the set of n × n matrices with entries in R and determinant in R^x, with group operation of matrix multiplication. The special linear group SL_n(R) is the set of n × n matrices with entries in R and determinant 1, with group operation of matrix multiplication.
    • Def 1.8: Let A be a set. Let A^-1={a^-1 | a ∈ a} and let B = A ∪ A^-1. Let B^* be the set of all words over B, including the empty word λ, and let ~ be the smallest equivalence relation on B satisfying (i) aa^-1 ~ λ ~ a^-1a for all a ∈ A and (ii) whenever p ~ q and x,y ∈ B^* then xpy ~ xqy. The free group on A is the set F(A) = B/~ of equivalence classes, with group operation [x][y]=[xy] for all x,y ∈ B.
  • Lemma 1.10: If G is a group, then:
    (1) There is only one element e ∈ G satisfying ae = a = ea for all a ∈ G (called the identity of G).
    (2) For each a ∈ G, there is only one element b ∈ G satisfying ab = e = ba (called the inverse of a and written a^-1).
  • Discussion of notation: aⁿ
  • Lemma 1.11: If G is a group and a,b,c, a₁,...,a_n ∈ G, then:
    (1) If ba = ca then b = c.
    (2) If ab = ac then b = c.
    (3) (a^-1)^-1 = a.
    (4) (a₁ ... a_n)^-1 = a_n^-1 ... a₁^-1.
    (5) (a^-1ba)ⁿ = a^-1bⁿa.

• Section B: Homomorphisms and isomorphisms
  • Def 1.14: A homomorphism from a group G to a group H is a function f:G → H satisfying f(gg') = f(g)f(g') for all g,g' ∈ G. An isomorphism from G to H is a homomorphism from G to H that is a bijection. Two groups G and H are isomorphic, written G ≅ H, if there is an isomorphism from G to H.
  • Prop 1.15: Suppose that φ: G → H is a homomorphism from a group G onto a group H.
    (1) φ(e_G) = e_H, where e_G and e_H are the identity elements of the groups G and H, respectively.
    (2) For every n ∈ Z, and a ∈ G, φ(aⁿ) = (φ(a))ⁿ.
    (3) For each a,b ∈ G, if ab = ba then φ(a)φ(b) = φ(b)φ(a).
  • Thm 1.16: "Isomorphic" is an equivalence relation on the set of all groups. Moreover, if φ:G → H and θ:H → K are isomorphisms, then φ^-1:H → G and θ ∘ φ:G → K are also isomorphisms.
  • Def 1.17: An Isomorphism Problem algorithm determines, upon input of two groups G and H, whether or not G ≅ H. A Classification Problem algorithm enumerates (lists) all of the groups up to isomorphism.
  • Rmk: There cannot be an IP or CP algorithm for all groups, but there are IP and CP algorithms for classes of "nice" groups.
  • Def 1.18: An isomorphism invariant is a property P of groups such that whenever G ≅ H and G has P then H has P.
  • Lemma 1.19: If P is an isomorphism invariant, G is a group that has P, and H is a group that does not have P, then G is not isomorphic to H.
  • Def 1.23: A group G is abelian (also called commutative) if for every a,b ∈ G, ab = ba.
  • Def 1.24: A group G is a finite group if the set G is finite.
  • Def 1.25: The order of a group G (denoted |G|) is the number of elements in the set G. The order of an element g of G (denoted |g|) is the smallest positive integer n such that gⁿ = e; if there is no such integer, then the order of g is infinite.
  • Examples
  • Def 1.26: A subset A of G is a generating set for G if every element of G is a (finite) product of elements of A and their inverses. (This is written G = ⟨ A ⟩.)
  • Def 1.27: A group G is finitely generated if there is a finite subset A of G that generates G.
  • Def 1.28: A group G is cyclic if there is an element a of G satisfying G = ⟨ {a} ⟩.
  • Thm 1.29: The following are isomorphism invariants.
    • The order of the group.
    • The set of orders of elements in the group.
    • "abelian".
    • "cyclic".
    • "finitely generated".
  • More examples:
    • Def 1.31: The quaternion group is the set Q₈ = {1,-1,i,-i,j,-j,k,-k} with group operation defined by 1a = a = a1 for all a in Q₈, (-1)a = -a = a(-1) and (-1)(-a) = a = (-a)(-1) for all a in {1,i,j,k}, ii = jj = kk = -1, ij = k, jk = i, ki = j, ji = -k, kj = -i, and ik = -j.
    • Def 1.32: A permutation of a set X is a bijection from X to X. The symmetric group on a set X, denoted S_X or Perm(X), is the group of all bijections from X to X (that is, the set of all permutations of X) with function composition. The symmetric group of degree n, denoted S_n, is Perm({1,2,...,n}).
  • Lemma 1.33: |S_n| = n!.
  • Examples
  • Def 1.36: An endomorphism of a group G is a homomorphism : G → G. An automorphism of a group G is an isomorphism : G → G.
  • Def 1.37: The automorphism group of a group G is the set Aut(G) of all automorphisms of G, with group operation of composition.
  • Prop 1.38: If G is a group, then Aut(G) is a group.
  • Examples

• Section C: Group actions: Definition and first examples
  • Def 1.40: A group action of a group G on a set X is a function G × X → X (written (g,x) → gx) satisfying:
    (1) g(g'x) = (gg')x for all g,g' ∈ G and x ∈ X, and
    (2) 1x = x for all x ∈ X.
  • Examples
    • The left action of G on G is defined by g ⋅ x = gx for all g,x ∈ G.
    • The conjugation action of G on G is defined by g ⋅ x = gxg^-1 for all g,x ∈ G.
  • Def 1.45: For a group G with a subset A of G that generates G, the Cayley graph Γ = Γ(G,A) has vertex set G and for each g ∈ G and a ∈ A, a directed edge from g to ga labeled a.
  • Prop 1.46: Let G be a group with a subset A that generates G. Then G acts on the Cayley graph Γ = Γ(G,A) by g(g') = (gg') for each element g ∈ G and vertex (g') of Γ, and g(g' ->- g'a) = (gg' ->- gg'a) for each element g ∈ G and edge (g' ->- g'a) of Γ.
  • Cor 1.47: Every group G is the group of symmetries of a metric space.
  • Lemma 1.49: Let G be a group and let X be a set.
    (1) If G acts on the set X (with action denoted by ⋅), then the function f:G → Perm(X) defined by (f(g))(x) := g ⋅ x (for all g in G and x in X) is a well-defined group homomorphism.
    (2) If f:G → Perm(X) is a homomorphism, then the function :G × X → X defined by g ⋅ x := (f(g))(x) (for all g in G and x in X) is a group action.
  • Def 1.50: Let G be a group acting on a set X. The equivalence relation on X induced by the action of G, written ~_G, is defined by p ~_G q if and only if there is a g ∈ G such that p = gq. The set of equivalence classes X/~_G is written X/G.
  • Lemma 1.51: Let G be a group acting on a set X. Then ~_G is an equivalence relation.
  • Def 1.52: Let G be a group acting on a set X, let p ∈ X, and let Y ⊆ X. The orbit of p is the equivalence class of p; that is, Orbit_G(p) = [p] = {gp | g ∈ G}. The stabilizer of p is Stab_G(p) = {g ∈ G | gp = p}, the pointwise stabilizer of Y is PtStab_G(Y) = {g ∈ G | gy = y for all y ∈ Y}, and the setwise stabilizer of Y is SetStab_G(Y) = {g ∈ G | gy ∈ Y for all y ∈ Y}.
  • Thm 1.53: (Orbit-Stabilizer Theorem): Let G be a finite group acting on a set X. If p ∈ X then |G| = |Orbit_G(p)| ⋅ |Stab_G(p)|.
  • Examples

Chapter 2: Group and homomorphism constructions: Subgroups and direct product groups

• Section A: Subgroups: Definition and examples
  • Def 2.1: A subgroup of a group G is a subset H of G that is a group under the operation of G. This is denoted H ≤ G; if H ≠ G this is also written H < G.
  • Lemma 2.2: A subset H of a group G is a subgroup of G if and only if H is nonempty and closed under multiplication and inversion.
  • Thm 2.3: (1) If H is a subgroup of G and K is a subgroup of H, then K is a subgroup of G. (2) If H_α is a subgroup of G for all α in an index set J, then ∩_{[α ∈ J]}H_α is a subgroup of G.
  • Examples:
    • Z < Q < R < C
    • Z^x < Q^x < R^x < C^x
    • Lemma 2.6: Let Y ⊆ X and let G be a subgroup of Perm(X). Then PtStab_G(Y) ≤ G and SetStab_G(Y) ≤ G.
    • Def 2.7: Let G be a group, and let G × G → G be the conjugation action of G on itself (that is, (g,h) → ghg^-1). Let H be a subset of G. The point-wise stabilizer PtStab_G(H) of H is called the centralizer of H in G, denoted C_G(H), and the set-wise stabilizer SetStab_G(H) of H is called the normalizer of H in G, denoted N_G(H). (As usual, if H = {p} for some element p of G, the centralizer is written C_G(p) and the normalizer is written N_G(p).)
    • Cor 2.8: For any subset H of a group G, C_G(H) ≤ G and N_G(H) ≤ G.
    • Lemma 2.9: If R ∈ {Z,Q,R,C,Z/mZ}, then SL_n(R) ≤ GL_n(R).
    • Def 2.10: Let G be a group. For each g ∈ G, the inner automorphism induced by g is the function σ_g,G=σ_g:G → G defined by σ_g,G(x) = gxg^-1 for all x ∈ G. The inner automorphism group of G is the subgroup Inn(G) = {σ_g,G | g ∈ G} of Aut(G).
  • Def 2.11: Let G be a group and let A be a subset of G. The subgroup of G generated by A, denoted ⟨ A ⟩, is the set of all products of elements of A and inverses of elements of A. A subgroup H of G is cyclic if H = ⟨ A ⟩ for a set A = {a} for some a ∈ G; this is denoted H = ⟨ a ⟩.
  • Prop 2.12: Let G be a group and let A be a subset of G. Then ⟨ A ⟩ is a subgroup of G. Moreover, ⟨ A ⟩ = ∩_{[A ⊆ H and H ≤ G]} H; that is, ⟨ A ⟩ is the intersection of all of the subgroups of G that contain A.

• Section B: Subgroups: Interactions with homomorphisms and group actions
  • Thm 2.15: Restrictions and extensions of homomorphisms are homomorphisms. That is: Let G and H be groups, let A be a subgroup of G, and let B be a subgroup of H.
    (1) The inclusion map i: A → G (defined by i(a) = a for all a ∈ A) is an injective homomorphism.
    (2) If f: G → H is a group homomorphism, then the restriction f|_A: A → H (defined by f|_A(a) = f(a) for all a ∈ A) is a homomorphism.
    (3) If f: G → H is a group homomorphism and f(G) ⊆ B, then the restriction f|^B: G → B (defined by f|^B(g) = f(g) for all g ∈ G) is a homomorphism.
    (4) If f: G → B is a group homomorphism, then the extension f|^H: G → H (defined by f|^H(g) = f(g) for all g ∈ G) is a homomorphism.
  • Thm 2.16: Homomorphic images and preimages of subgroups are subgroups. That is: Let G and H be groups and let f: G → H be a homomorphism.
    (1) If A is a subgroup of G then f(A) is a subgroup of H.
    (2) If B is a subgroup of H then f^-1(B) is a subgroup of G.
  • Def 2.17: Let f: G → H be a group homomorphism. The kernel of f is the subgroup Ker(f) = f^-1({1_H}) = {g ∈ G | f(g) = 1_H} of G. The image of f is the subgroup Im(f) = f(G) of H.
  • Def 2.18: An embedding of a group G in a group H is an injective homomorphism :G → H.
  • Prop 2.19: If f: G → H is an embedding, then G ≅ f(G) ≤ H.
  • Thm 2.20: (Cayley's Thm): Each group G embeds in Perm(G), and hence each finite group G embeds in S_|G|.
  • More examples:
    • Lemma 2.22: Let B ⊆ A. Then F(B) embeds in F(A) and Perm(B) embeds in Perm(A).
    • Lemma 2.23: If m ≤ n then S_m embeds in S_n.
  • Thm 2.26: If G is a group acting on a set X and H is a subgroup of G, then the restriction to H of the action of G is an action of H on X. Moreover, each G-orbit [x]_G is a union [x]_G = ∪_{p ∈ [x]G} [p]_H of H-orbits.

• Section C: Subgroups: Interactions with isomorphism invariants
  • Order of the group:
    • Thm 2.27: Every subgroup of a finite group is finite. (There exist infinite groups with finite subgroups.)
    • Thm 2.28: (Lagrange's Theorem): If H is a subgroup of a finite group G, then |H| divides |G|.
  • Orders of elements:
    • Thm 2.30: If H ≤ G, then the set of orders of elements of H is a subset of the set of orders of elements of G.
  • Abelian:
    • Thm 2.32: Every subgroup of an abelian group is abelian. (There exist nonabelian groups with abelian subgroups.)
    • Def 2.33: The center of a group G is Z(G) = {g ∈ G | gh = hg for all h ∈ G}.
    • Thm 2.34: (1) If G is a group, then Z(G) is an abelian subgroup of G. (2) The order of the center of a group is an isomorphism invariant.
  • Cyclic:
    • Thm 2.36: Every subgroup of a cyclic group is cyclic. (There exist noncyclic groups with cyclic subgroups.)
    • Prop 2.37: Let G be a cyclic group generated by an element g of finite order n.
      (a) |⟨ g ⟩| = |g|.
      (b) If k is an integer, then |g^k| = n/gcd(k,n).
      (c) There is a bijection ψ : {divisors of |G|} → {subgroups of G} given by ψ(d) = ⟨ g^|G|/d ⟩ for each divisor d of |G|; moreover, for each subgroup H of G, ψ^-1(H) = |H|.
    • Thm 2.39: (Homomorphism Building Theorem (HBT) for Cyclic Groups): Suppose that G = ⟨ a ⟩ and H are groups. (a) If |a| = ∞, then for every y in H there is a unique group homomorphism f: G → H such that f(a) = y. (b) If |a| = n < ∞, then for every y in H satisfying yⁿ = e_H, there is a unique group homomorphism f: G → H such that f(a) = y.
    • Def 2.40: The infinite cyclic group is the group C_∞ = {aⁱ | i ∈ Z} with multiplication aⁱa^j = a^i+j. For any natural number n, the cyclic group of order n is the group C_n = {aⁱ | i ∈ {0,...,n-1}} with multiplication aⁱa^j = a^{i+j(mod n)}.
    • Thm 2.41: (Classification Theorem for Cyclic Groups): The list of isomorphism classes of cyclic groups is [C_∞] together with [C_n] for all n ∈ N.
  • Finitely generated:
    • Thm 2.44: There exist a finitely generated group G and a subgroup H of G such that H is not finitely generated. (There exist a non-finitely-generated group G and a subgroup H of G with H finitely generated.)

• Section D: Direct product groups: Definitions and examples
  • Def 2.50: (a) Let G_i be a group for all 1 ≤ i ≤ n. The direct product of the groups G_i is the Cartesian product G₁ × G₂ × ... × G_n with multiplication defined by (g₁,...,g_n)(h₁,...,h_n) = (g₁h₁,...,g_nh_n).
    (b) Let G_α be a group for all α in an index set J. The direct product of the groups G_α is the Cartesian product Π_{[α ∈ J]} G_α with multiplication defined by (g_α)_{α ∈ J}(h_α)_{α ∈ J} = (g_αh_α)_{α ∈ J}. The direct sum of the groups G_α is the subset ⊕_{[α ∈ J]} G_α of the direct product Π_{[α ∈ J]} G_α given by ⊕_{[α ∈ J]} G_α = {(g_α)_{α ∈ J} | g_α = 1_Gα for all but finitely many α}, with the same multiplication as the direct product.
  • Thm 2.51: The direct product of a collection of groups is a group, and the direct sum of the collection is a subgroup of the direct product.
  • Examples

• Section E: Direct product groups: Interactions with homomorphisms and group actions
  • Thm 2.54: Let Π_{[α ∈ J]} G_α be a direct product of groups. For each β in J, the projection p_β: Π_{[α ∈ J]} G_α → G_β is a homomorphism.
  • Thm 2.55: For each index β ∈ J, the inclusion function i_β:G_β → Π_{[α ∈ J]} G_α, defined by i(g) = (g_α)_{α ∈ J} where g_β = g and g_α = 1_Gα for all α ≠ β, is an embedding.
  • Thm 2.57: Let H be a group and let f_β:H → G_β be a function for all β ∈ J. Then the product function f:H → Π_{[α ∈ J]} G_α defined by f(h) = (f_α(h))_{α ∈ J} for all h ∈ H (that is, f_β = p_β ∘ f for all β ∈ J) is a group homomorphism if and only if f_β is a group homomorphism for all β ∈ J.
  • Prop 2.59: (1) If G_α ≅ H_α for all α ∈ J, then Π_{[α ∈ J]} G_α ≅ Π_{[α ∈ J]} H_α. (2) G × H ≅ H × G.
  • Thm 2.61: If G acts on a set X and H acts on a set Y, then G × H acts on the set X × Y by (g,h) ⋅ (x,y) = (g ⋅ x,h ⋅ y) for all g ∈ G, h ∈ H, x ∈ X, and y ∈ Y.

• Section F: Direct product groups: Interactions with group constructions
  • Thm 2.66: If H_α is a subgroup of G_α for all α ∈ J, then Π_{[α ∈ J]} H_α is a subgroup of Π_{[α ∈ J]} G_α.
  • Examples

• Section G: Direct product groups: Interactions with isomorphism invariants
  • Order of the group:
    • Thm 2.70: A direct product of finitely many finite groups is finite. A direct product of infinite groups is infinite.
  • Orders of elements:
    • Thm 2.73: The set of orders of elements of a direct product Π_{[α ∈ J]} G_α contains the union of the sets of orders of elements of the factor groups G_α.
    • Thm 2.74: (1) Let g = (g_α)_{α ∈ J} be an element of a direct product group Π_{[α ∈ J]} G_α. If there is an index β ∈ J such that |g_β| = ∞, then |g| = ∞.
      (2) Let g = (g₁,...,g_n) be an element of a direct product Π_i=1ⁿ G_i. If |g_i| < ∞ for all i, then |g| = lcm{|g₁|,...,|g_n|}.
  • Abelian:
    • Thm 2.77: Every direct product of abelian groups is abelian.
  • Cyclic:
    • Thm 2.80: Every direct product of noncyclic groups is noncyclic. There exists a direct product of cyclic groups that is not cyclic.
  • Finitely generated:
    • Thm 2.83: Every direct product of finitely many finitely generated groups is finitely generated. A direct product of infinitely many nontrivial groups is not finitely generated.

Chapter 3: Group and homomorphism constructions: Quotient groups

• Section A: Definitions and examples
  • Def 3.1: An equivalence relation ~ on a group G is compatible with multiplication if whenever g,h,k ∈ G and g ~ h then gk ~ hk and kg ~ kh.
  • Def 3.2: Let G be a group and let ~ be an equivalence relation on G that is compatible with multiplication. The quotient group is the set G/~ of equivalence classes, with group multiplication [g][h] = [gh].
  • Lemma 3.3: Let G be a group, let ~ be an equivalence relation on G, let G/~ be the set of equivalence classes, and define [g][h] = [gh]. Then G/~ with this operation is a (well-defined) group if and only if ~ is compatible with multiplication.
  • Lemma 3.6: Let N be a subgroup of a group G. The equivalence relation ~_N on G induced by the left action of N is given by g ~_N h if and only if h = ng for some n ∈ N. The equivalence class of g ∈ G (also called the orbit of g) is Ng = {ng | n ∈ N}, called the right coset of N in G containing g.
  • Def 3.7: A subgroup N of a group G is normal in G, written N ⊴ G (and if N ≠ G, this is written N ⊲ G), if gNg^-1 = N for all g ∈ G.
  • Prop 3.8: Let N be a subgroup of a group G. The following are equivalent: (1) N ⊴ G. (2) The equivalence relation ~_N induced by the left action of N on G is compatible with multiplication. (3) gNg^-1 ⊆ N for all g ∈ G. (4) N_G(N) = G. (5) gN = Ng for all g ∈ G.
  • Prop 3.10: Let G be a group. An equivalence relation ~ on G is compatible with multiplication if and only if ~ = ~_N for some normal subgroup N of G.
  • Notation 3.12: For any normal subgroup N of a group G, the quotient group G/~_N is denoted G/N. (For any equivalence relation on a group G that is compatible with multiplication, the quotient group G/~ is denoted G/N, where N = N_~ := {g ∈ G | G ~ e}.)
  • Examples
    • {1},G ⊴ G and G/{1} ≅ G, G/G ≅ {1}.
    • The infinite dihedral group D_∞ is the set D_∞ = {rⁱ,rⁱs | i ∈ Z} with multiplication defined by (rⁱ)(r^j) = r^i+j, (rⁱ)(r^js) = r^i+js, (rⁱs)(r^j) = r^i-js, and (rⁱs)(r^js) = r^i-j. Then ⟨ rⁿ ⟩ ⊴ G and G/⟨ rⁿ ⟩ ≅ D_2n.
  • Lemma 3.14: Let G be a group, let H be a subgroup of G, and let p,q ∈ G. The following are equivalent: (1) There is a g ∈ G with p,q ∈ gH. (2) pH = qH. (3) p = qh for some h ∈ H. (4) p^-1q ∈ H. (5) q^-1p ∈ H.
  • Lemma 3.15: All subgroups of abelian groups are normal.
  • Thm 3.16: Let H = ⟨ S ⟩ be a subgroup of G = ⟨ T ⟩. Then H is normal in G if and only if usu^-1 ∈ H for all s ∈ S and u ∈ T ∪ T^-1.

• Section B: Interactions with homomorphisms and group actions
  • Thm 3.20: Let N ⊴ G. The quotient map q:G → G/N defined by q(g) = gN is a group homomorphism with kernel Ker(q) = N.
  • Cor 3.21: A subgroup N of a group G is normal in G if and only if N is the kernel of a homomorphism with domain G.
  • Prop 3.25: (Homomorphism Building Theorem (HBT) for Quotient Groups): Let N be a normal subgroup of a group G and let f: G → H be a homomorphism satisfying the property that N ⊆ Ker(f). Then:
    (1) There is a well-defined homomorphism φ:G/N → H defined by φ(gN) = f(g) for all gN in G/N.
    (2) If f is onto, then φ is onto.
    (3) If f also satisfies the property that Ker(f) ⊆ N, then φ is one-to-one.
  • Cor 3.26: (First Isomorphism Theorem (1IT)): If f: G → H is a homomorphism, then Ker(f) ⊴ G and G/Ker(f) ≅ f(G).
  • Def 3.29: For a subgroup H in G, the index of H in G, denoted |G : H|, is the number of left cosets of H in G.
  • Cor 3.30: If f:G → H is a homomorphism, then (1) f is injective if and only if Ker(f) = {1}, and (2) |G : Ker(f)| = |f(G)|.
  • Cor 3.31: (Lagrange's Theorem): If H is a subgroup of a finite group G, then |G| = |H| |G:H|.
  • Examples
  • Thm 3.35: (Homomorphism Building Theorem (HBT) for Free Groups): Let A be a set, let F(A) be the free group on A, let H be a group, and let j:A → H be a function. Then there is a unique homomorphism f:F(A) → H satisfying f([a]) = j(a) for all a ∈ A.
  • Cor 3.36: Every group is a quotient of a free group. Every finitely generated group is a quotient of a finitely generated free group.
  • Def 3.40: Let G be a group and let A ⊆ G. The normal subgroup of G generated by A, denoted ⟨ A ⟩^N, is the set of all products of conjugates of elements of A and inverses of elements of A. In symbols, ⟨ A ⟩^N = { g₁a₁^i₁g₁^-1 ... g_ma_m^i_mg_m^-1 | m ≥ 0, and each a_j ∈ A, g_j ∈ G, i_j ∈ {1,-1}}.
  • Prop 3.41: Let G be a group and let A be a subset of G. Then ⟨ A ⟩^N is a normal subgroup of G. Moreover, ⟨ A ⟩^N = ∩_{[A ⊆ H and H ⊴ G]} H; that is, ⟨ A ⟩^N is the intersection of all of the normal subgroups of G that contain A.
  • Def 3.44: Let A be a set and let R be a subset of the free group F(A). The quotient group G = F(A)/⟨ R ⟩^N has presentation ⟨ A | R ⟩ = ⟨ A | {r = 1 | r ∈ R} ⟩.
  • Examples
  • Thm 3.45: (Homomorphism Building Theorem (HBT) for Presentations): Let A be a set, let F(A) be the free group on A, let R be a subset of F(A), let H be a group, and let j:A → H be a function satisfying the property that whenever r = a₁^i₁ ... a_m^i_m ∈ R (with each a_j ∈ A, g_j ∈ G and i_j ∈ {1,-1}) then (j(a₁))^i₁ ... (j(a_m))^i_m = 1_H. Then there is a unique homomorphism f: ⟨ A | R ⟩ → H satisfying f([a]) = j(a) for all a ∈ A.
  • Examples
  • Rmk: There is no algorithm that, upon input of a finite presentation of a group G, can determine whether G = {1}.

• Section C: Interactions with isomorphism invariants
  • Def 3.48: A group G is called an extension of a group N by a group H if N ⊴ G and G/N ≅ H.
  • Order of the group:
    • Thm 3.50: Every quotient of a finite group is finite. Moreover, if G is a finite group and G/N is a quotient of G, then |G/N| divides |G|.
    • Thm 3.51: Every extension of finite groups is finite. That is: If N ⊴ G and both N and G/N are finite groups, then G is finite.
  • Orders of elements:
    • Prop 3.52: If g is an element of a group G and N ⊴ G, then |g| ≥ |gN|. Moreover, if g has finite order, then |gN| divides |g|.
  • Abelian:
    • Thm 3.55: Every quotient of an abelian group is abelian.
    • Def 3.57: Let G be a group. For any g,h ∈ G, the commutator of g with h is the element [g,h] = ghg^-1h^-1 of G. The commutator subgroup of G is the group G' = ⟨ {ghg^-1h^-1 | g,h ∈ G} ⟩. The abelianization of G is the quotient group G_ab = G/G'.
    • Prop 3.58: For any group G, G' ⊴ G and G_ab is the largest abelian quotient of G.
    • Prop 3.59: The isomorphism class of the abelianization (and hence the order of the abelianization, whether the abelianization is cyclic, etc.) is an isomorphism invariant.
  • Cyclic:
    • Thm 3.60: Every quotient of a cyclic group is cyclic.
  • Finitely generated:
    • Thm 3.63: Every quotient of a finitely generated group is finitely generated.
    • Thm 3.65: Every extension of finitely generated groups is finitely generated. That is: If N ⊴ G and both N and G/N are finitely generated groups, then G is finitely generated.

• Section D: Interactions with group constructions
  • Examples
  • Prop 3.71: Let G be a group, let K, N ⊴ G. Then KN, K ∩ N ⊴ G.
  • Thm 3.73: (Second Isomorphism Theorem (2IT)): Let G be a group, let H ≤ G and let N ⊴ G. Then H ∩ N ⊴ H, HN ≤ G, N ⊴ HN, and there is an isomorphism H/(H ∩ N) ≅ HN/N given by h (H ∩ N) → hN.
  • Thm 3.77: (Third Isomorphism Theorem (3IT)): Let G be a group and let H,K ⊴ G with H ≤ K. Then H ⊴ K and K/H ⊴ G/H, and there is an isomorphism (G/H)/(K/H) ≅ G/K given by gH(K/H) → gK.
  • Lemma 3.79: Homomorphic images (sort of) and preimages of normal subgroups are normal subgroups. That is: Let G and H be groups and let f: G → H be a homomorphism.
    (1) If A is a normal subgroup of G then f(A) is a normal subgroup of f(G).
    (2) If B is a normal subgroup of H then f^-1(B) is a normal subgroup of G.
  • Thm 3.80: (Lattice Isomorphism Theorem (LIT)): Let G be a group, let N be a normal subgroup in G, and let q: G → G/N be the quotient map. Then the function Ψ : {subgroups of G containing N} → {subgroups of G/N} defined by Ψ(H) = q(H) = H/N is a bijection with inverse defined by Ψ^-1(M) = q^-1(M) for each M ≤ G/N. Moreover, Ψ and Ψ^-1 preserve subgroup, index, generation, intersection, and normal subgroup structure.

• Section E: Decomposing groups: Subnormal series
  • Def 3.82: Let G be a group. A subnormal series for G is a sequence of subgroups {e}=N₀ ⊴ N₁ ⊴ ... ⊴ N_k = G for some k ≥ 0 such that for all i ∈ {0,...,k-1} we have N_i ⊴ N_i+1 ≤ G. The integer k is the length of the series, and the quotient groups N_i+1/N_i are the factor groups.
  • Def 3.83: Let P be an isomorphism invariant property. A group G is poly-P if G has a subnormal series in which each factor group has the property P.
  • Prop 3.84: If P is an isomorphism invariant property, then so is poly-P.
  • Def 3.85: A group G is solvable if G is poly-abelian.
  • Examples
    • Thm 3.87: Let n be a natural number.
      (1) Every element of S_n is a product of disjoint cycles.
      (2) An m-cycle in S_n has order m and is a product of m-1 transpositions.
      (3) Every element of S_n is a product of either an even number or an odd number of transpositions, but not both.
      (4) The subgroup of S_n of products of even numbers of transpositions, called the alternating group of degree n and denoted A_n, has index |S_n : A_n| = 2 and A_n ⊴ S_n.
      (5) A_n is not abelian for all n ≥ 4.
    • Lemma 3.88: S₄ is polycyclic and solvable.
    • Thm 3.89: If G is a polycyclic group, then G is solvable. The group ⊕_{i ∈ Z} Z/2Z is solvable but not polycyclic.
  • Thm 3.90: Every subgroup and quotient of a solvable group is solvable.
  • Thm 3.91: (1) Every extension and direct product of solvable groups is solvable.
    (2) Let P be an isomorphism invariant property. Every extension and direct product of poly-P groups is poly-P.
  • Thm 3.92: Poly-poly-P = poly-P and hence poly-solvable = solvable.
  • Def 3.94: A group H is simple if |H| > 1 and the only normal subgroups of H are {e} and H.
  • Def 3.95: A composition series for a group G is a subnormal series such that every factor group is simple.
  • Thm 3.96: (Jordan-Holder Theorem): Every finite group has a composition series. Moreover, if {e}=M₀ ⊴ M₁ ⊴ ... ⊴ M_j = G and {e}=N₀ ⊴ N₁ ⊴ ... ⊴ N_k = G are two composition series for G, then j = k and there is a permutation s ∈ S_k such that M_i/M_i-1 ≅ N_s(i)/N_s(i)-1 for all 1 ≤ i ≤ j.
  • Thm 3.98: Let G be a finite group. The following are equivalent: (1) G is solvable. (2) G is polycyclic. (3) G is poly-(cyclic of prime order).

Chapter 4: Applications of group actions and automorphisms

• Section A: Representations and applications
  • Def 4.1: A permutation representation of a group G is a homomorphism f: G → Perm(X) for some set X. A linear representation of G is a homomorphism G → GL_n(R) for some natural number n and field R. A representation f is faithful if Kerf(f) = 1.
  • Rmk: Note GL_n(R) ≤ Perm(Rⁿ). Eg R = Q, R, C, Z/pZ for a prime p.
  • Def 4.3: For a group G acting on a set X, the induced homomorphism f: G → Perm(X) is the permutation representation induced by the action. The action is faithful if f is faithful. The action is transitive if for all p,q ∈ X there is a g ∈ G such that q=gp.
  • Thm 4.4: (LOIS): Let G be a group acting on a set X and let p ∈ X. Then Stab_G(p) is a subgroup of G and |Orbit_G(p)| = |G : Stab_G(p)|.
  • Groups acting on themselves and their cosets by left multiplication:
    • Thm 4.6: Let H be a subgroup of a group H. Then G acts transitively on the set G/H of left cosets by left multiplication (g · (g'H) = (gg')H), and Stab_G(1_G/H) = H. If π is the induced permutation representation, then Ker(π) = ∩_{g ∈ G} gHg^-1 is the largest subgroup of G containing H.
    • Def 4.7: The group action in Thm 4.6 is the left regular representation of G over H.
    • Thm 4.8: If G is a finite group of order n and p is the smallest prime dividing |G|, then any subgroup H of G of index p is normal in G.
  • Groups acting on themselves and their subsets by conjugation:
    • Def 4.10: Let G be a group. Two elements g,g' ∈ G are conjugate if there is an h ∈ G with g' = hgh^-1. The conjugacy class of an element g ∈ G is [g]_c = {hgh^-1 | h ∈ G}. Two subsets S,S' ⊆ G are conjugate if there is an h ∈ G with S' = hSh^-1.
    • Thm 4.11: Let G be a group.
      (1)Then G acts on G by conjugation (g · g' = gg'g^-1). For all g ∈ G, the orbit of g is the conjugacy class of g, and |[g]_c| = |G : C_G(g)|.
      (2) Then G acts on the power set P(G) by conjugation (g · S = gSg^-1). For all S ∈ P(G), |Orbit_G(S)| = |G : N_G(S)|.
    • Cor 4.13: For a finite group G and element g ∈ G, |[g]_c| divides |G|.
    • Thm 4.15: (Class equation): Let G be a finite group and let g₁,...,g_r ∈ G be a list of unique representatives of all of the conjugacy classes of G of size greater than 1. Then |G| = |Z(G)| + ∑_i=1^r |G : C_G(g_i)|.
    • Thm 4.17: If p is a prime number and G is a finite group of order p^m for some m > 0, then Z(G) is not the trivial group.
    • Def 4.18: For a prime number p (≥ 2), a p-group is a group of order p^m for some m > 0.
    • Cor 4.19: If G is a p-group for some prime number p, then G is solvable.
    • Thm 4.20: If G/Z(G) is cyclic, then G is abelian.
    • Examples
      • Def 4.22: Let t be an element of S_n. The cycle type of t is the set of numbers {m₁,...,m_n} such that the disjoint cycle decomposition of t consists of n cycles of lengths m₁,...,m_n.
      • Thm 4.23: Let s,t ∈ S_n. (1) The elements s and t are conjugate in S_n if and only if they have the same cycle type. (2) Let (a_1,1 ... a_1,m1) ... (a_n,1 ... a_n,mn) be the disjoint cycle decomposition of s. Then tst^-1 has disjoint cycle decomposition (t(a_1,1) ... t(a_1,m1)) ... (t(a_n,1) ... t(a_n,mn)).
      • Prop 4.24: Let G be a group and H ≤ G. Then H ⊴ G if and only if H is a union of conjugacy classes of G.
      • Thm 4.25: The alternating group A_n is simple for all n ≥ 5.

• Section B: Automorphisms and semidirect products
  • Thm 4.28: Let G be a group. Then: (1) Inn(G) ⊴ Aut(G). (2) For each g ∈ G and φ ∈ Aut(G), φ σ_g φ^-1 = σ_φ(g) (where σ_g denotes the inner automorphism induced by g).
  • Def 4.29: Let G be a group. The outer automorphism group of G is Out(G) = Aut(G)/Inn(G).
  • Thm 4.30: Inn(G) ≅ G/Z(G).
  • Examples
    • Prop 4.32: For any natural number n, Aut(C_n) ≅ Out(C_n) ≅ (Z/nZ)^x and Inn(C_n) ≅ 1.
  • Thm 4.34: If H ⊴ G, then G/C_G(H) embeds in Aut(H).
  • Def 4.40: Let N and K be groups and let φ: K → Aut(N) be a homomorphism. The semidirect product induced by φ is the set N × K with the binary operation defined by (n,k)(n',k') = (n((φ(k))(n')),kk'); this group is denoted by N ⋊_φ K.
  • Thm 4.41: If N and K are groups and φ: K → Aut(N) is a homomorphism, then: (1) N ⋊_φ K is a group. (2) N ≅ N' = {(n,1_K) | n ∈ N} ⊴ N ⋊_φ K and K ≅ {(1_N,k) | k ∈ K} ≤ N ⋊_φ K. (3) N ⋊_φ K / N' ≅ K.
  • Examples
  • Thm 4.42: N ⋊_φ K = N × K if and only if φ is the trivial homomorphism (that is, φ(k) = 1_Aut(N) for all k ∈ K).
  • Thm 4.44: Let N = ⟨ x | x^m = 1 ⟩ and M = ⟨ y | yⁿ = 1 ⟩, and let 1 ≤ j ≤ m-1 satisfy gcd(j,m) = 1 and jⁿ = 1. Then there is a unique homomorphism φ: K → Aut(N) satisfying (φ(y))(x) = x^j, and N ⋊_φ K ≅ ⟨ r,s | r^m = 1, sⁿ = 1, srs^-1 = r^j ⟩.
  • Thm 4.47: Let G be a group with subgroups N and K satisfying N ⊴ G, G = NK, and N ∩ K = 1. Then: (1) G ≅ N ⋊_φ K where φ: K → Aut(N) is defined by φ(k)(n) = knk^-1 for all k ∈ K and n ∈ N. (2) If K ⊴ G also, then G ≅ N × K.
  • Def 4.48: If N ⊴ G, K ≤ G, G = NK, and N ∩ K = 1, then G is an internal semidirect product of N by K. If K ⊴ G as well, then G is the internal direct product of N and K.
  • Thm 4.49: If N = ⟨ A | R ⟩, K = ⟨ B | S ⟩, and φ: K → Aut(N) is a homomorphism, then N ⋊_φ K ≅ ⟨ ⟨ A ∪ B | R ∪ S ∪ {φ(k)(n) = knk^-1 | k ∈ K and n ∈ N} ⟩.

• Section C: Classifying finite groups, and Sylow's Theorem(s)
  • Applications of group actions and semidirect products
    • Thm 4.50: If G is a group of order p with p prime, then G is cyclic.
    • Thm 4.51: (Cauchy's Theorem) If G is a finite group and p is a prime number dividing |G|, then G has an element of order p. (In fact, at least p-1 elements of order p.)
    • Lemma 4.52: (1) If n ∈ N, then Aut(Z/nZ) ≅ (Z/nZ)^x. Hence if q is prime, then |Aut(Z/qZ)| = q-1.
      (2) If n ∈ N, then C_n² ≇ C_n × C_n.
      (3) If m,n ∈ N and gcd(m,n) = 1, then C_mn ≅ C_m × C_n.
      (4) If M is a finite group of order m, N is a finite group of order n, and gcd(m,n) = 1, then Aut(M × N) ≅ Aut(M) × Aut(N).
      (5) If p is a prime and k ∈ N, then Aut((Z/pZ)^k) ≅ GL_k(Z/pZ). [Here (Z/pZ)^k denotes the direct product (Z/pZ) × ... × (Z/pZ) of k copies of Z/pZ.]
    • Cor 4.53: If G is a group of order pq with p,q prime, p ≤ q, and p does not divide q-1, then G is abelian.
    • Cor 4.54: Let p and q be prime numbers with p < q.
      (1) If G is a group of order pq, then G is isomorphic to C_q ⋊_φ C_p for some homomorphism φ: C_p → Aut(C_q).
      (2) If p ∤ (q-1), then the only group of order pq is the cyclic group C_pq, up to isomorphism.
      (3) If p | (q-1), then there is a nonabelian group of order pq.
      (4) If p | (q-1), then there are exactly two groups of order pq, up to isomorphism.
    • Examples
    • Cor 4.56: If p is a prime number, then every group of order p² is isomorphic to exactly one of C_p² or C_p × C_p.
  • Sylow's main theorem
    • Def 4.60: Let p be a prime and let G be a finite group of order pⁿm where p ∤ m. A Sylow p-subgroup of G is a subgroup of G of order pⁿ. Let Syl_p(G) denote the set of Sylow p-subgroups of G. Let n_p(G) = |Syl_p(G)|.
    • Thm 4.63: (Sylow's Main Theorem): Let G be a finite group of order pⁿm where p is a prime and p ∤ m.
      (1) Syl_p(G) ≠ ∅.
      (2) If Q ∈ Syl_p(G) and R ≤ G is any p-subgroup of G, then there is a g ∈ G such that R ≤ gQg^-1.
      (3) |Syl_p(G)| ≡ 1 (mod p).
      (4) If Q ∈ Syl_p(G), then |G : N_G(Q)| = |Syl_p(G)|, and hence |Syl_p(G)| | m.
    • Example: D₆
    • Lemma 4.65: Let G be a group.
      (1) If m ∈ N and there is exactly one subgroup H of G of order m, then H ⊴ G.
      (2) If p is a prime number, then a Sylow p-subgroup of G is normal if and only if |Syl_p(G)| = 1.
    • Prop 4.67: Suppose that G is a finite group, N ⊴ G, K ≤ G, n = |N|, m = |K|, and gcd(m,n) = 1. Then NK is a subgroup of G of order mn, and NK is an internal semidirect product of N by K. Consequently, if |G| = mn, then G = NK is an internal semidirect product of N by K.
    • Example: Classification of groups of order 45
    • Thm 4.69: If p is a prime number and G is a finite group of order pm with p ∤ m, then G contains (p-1)|Syl_p(G)| elements of order p.
    • Thm 4.70: Let K be a finite cyclic group and let N be arbitrary group. Suppose that φ: K → Aut(N) and θ: K → Aut(N) are homomorphisms satisfying the property that there is a σ ∈ Aut(N) such that σ φ(K) σ^-1 = θ(K); that is, the images of φ and θ are conjugate subgroups of Aut(N). Then N ⋊_φ K ≅ N ⋊_θ K.
    • Prop 4.71: Let p be a prime number, let n be a natural number, and let (Z/pZ)ⁿ = Z/pZ × ... × Z/pZ be the direct product of n copies of Z/pZ. Then Aut((Z/pZ)ⁿ) ≅ GL_n(Z/pZ) is a group of order ∏_i=0^n-1 (pⁿ - pⁱ).
    • Prop 4.72: Let K and N be groups and let φ,θ: K → Aut(N) be homomorphisms. If there is an automorphism β ∈ Aut(K) such that θ = φ ∘ β, then N ⋊_φ K ≅ N ⋊_θ K.
    • More examples: Classification of groups of order 30, classification of simple groups of order 48, classification of groups of order 75.
  • Rmk 4.75: Flow chart/strategies for classifying all finite groups of a given order m:
    • (1) Factor m = p₁^a₁ ... p_s^a_s with each p_i prime, p₁ < ... < p_r, and each a_i ∈ N.
    • (2) For each prime p_i use Sylow Theorem 4.63(4) to write down a set U_i of natural numbers containing n_pi(G) = |Syl_pi(G)| (namely the set of divisors of m/(p_i^a_i)), and use Sylow Theorem 4.63(3) to remove some of the elements of U_i that are not equal to n_pi(G).
    • (3) For all indices 1 ≤ i ≤ r for which a_i = 1, use Thm 4.69 to count the number of elements of order p_i in G. For the other indices i, use Cauchy's theorem to get a lower bound on the number of elements in G of order p_i. Also consider (sub)cases of potential values of the n_pi(G) and what they imply about the existence of elements of order p_i^j for various j ∈ N, or alternatively consider (sub)cases of potential values of orders of elements of G, and what they imply about the values of the n_pi(G), and/or the isomorphism class of the Sylow p-subgroups of G. Use this information to remove more elements of U_i that are not equal to n_pi(G).
    • (4) Try to show that G is an extension of smaller nontrivial groups N and G/N (that is, G is not simple) by:
      • (4a) For an index i with n_pi(G) = 1, TOC Lemma 4.65 says that there is a normal Sylow p_i-subgroup.
      • (4b) For an index i with n_pi(G) ≠ 1, the group G acts on the set Syl_pi(G) by conjugation; the kernel of the permutation representation π: G → Perm(Syl_pi(G)) is a normal subgroup of G.
      • (4b') For any index i, use Thm 4.17 to find a nontrivial normal subgroup (the center) of each Sylow p_i-subgroup.
      • (4c) Use products of Sylow subgroups with normal subgroups and Prop 4.67 to build up larger subgroups in G, and then use Lagrange's Theorem and Thm 4.8 to find normal subgroups.
      Then classify the smaller groups N and G/N using the techniques in this flow chart (and Thms 4.50, 4.54, 4.56), and use these classifications to determine information about G.
    • (5) Try to show that every group G of order m is an internal semidirect product of a normal subgroup N of order s by a subgroup K of order t. Then:
      • (5a) Classify all groups of orders s and t using the techniques in this flow chart (and Thms 4.50, 4.54, 4.56); let S be the set of all groups of order s and let T be the set of all groups of order t, up to isomorphism.
      • (5b) Using TOC Lemma 4.52, analyze Aut(N) for each N ∈ S. Then for each N ∈ S and K ∈ T find the set W(K,N) of all homomorphisms K → Aut(N).
      • (5c) For each pair of 3-tuples (N,K,φ),(N',K',φ') with N,N' ∈ S, K,K' ∈ T, φ ∈ W(K,N), and φ' ∈ W(K',N'), determine whether or not N ⋊_φ K ≅ N' ⋊_φ' K'.
        • (5ci) To show that N ⋊_φ K ≅ N' ⋊_φ' K', try to use Thm 4.70 or Prop 4.72.
        • (5cii) To show that N ⋊_φ K ≇ N' ⋊_φ' K', try to show that the abelianizations of these two groups are not isomorphic.
  • Proof of Sylow's Main Theorem
    • Lemma 4.80: Let G be a finite group, p a prime, Q a Sylow p-subgroup of G, and R any p-subgroup of G. Then R ∩ N_G(Q) = R ∩ Q.
  • Classification of finitely generated abelian groups
    • Thm 4.85: Let G be a finite abelian group and factor the order m of G as m = p₁^a₁ ... p_s^a_s such that s ≥ 0, each p_i is prime, p₁ < ... < p_s, and each a_i ∈ N. Then
      (1) G ≅ Q₁ × ... × Q_s where |Q_i| = p_i^a_i for all i.
      (2) For each index i, there is a partition a_i = a_i,1 + ... + a_i,ji with each a_i,j ≥ 1, such that Q_i ≅ (Z/p_i^a_i,1Z) × ... × (Z/p_i^a_i,jiZ).
      (3) The p_i's, j_i's and a_i,j's are uniquely determined by G.
    • Def 4.86: In Thm 4.80, the p_i^a_i are the elementary divisors of G, and the decomposition of G in Thm 4.80(1-2) is the elementary divisor decomposition of G.
    • Def 4.87: The direct product Z^r = Z × ... × Z of r copies of Z is the free abelian group of rank r.
    • Thm 4.88: (Classification of finitely generated abelian groups (CFGAG)): Let G be a finitely generated abelian group. Then
      (1) G ≅ Z^r × (Z/n₁Z) × ... × (Z/n_sZ) for some r ≥ 0, s ≥ 0, and n_i ≥ 2 for all i, satisfying n_i+1 | n_i for all i.
      (2) The integers r,s,n₁,...,n_s are uniquely determined by G.
    • Def 4.89: In Thm 4.84, the number r is the rank of G, the numbers n₁,...,n_s are the invariant factors of G, and the decomposition of G in Thm 4.84(1) is the invariant factor decomposition of G.

Chapter 5: Introduction to rings

• Section A: Definitions and first examples
  • Motivation: Studying matrices (linear algebra), polynomials, and Z,Q,R,C, and proving things about them all at once instead of individually.
  • Def 5.1: A monoid is a set S together with a binary operation ⋅ that is associative and has an identity. A semigroup is a set S together with a binary operation ⋅ that is associative.
  • Def 5.2: A ring is a set R together with two binary operations + and ⋅ satisfying the following:
    (1) (R,+) is an abelian group (with identity element denoted 0).
    (2) (R,⋅) is a semigroup.
    (3) (R,+,⋅) is distributive: For all a,b,c ∈ R, a(b + c) = ab + ac.
  • Def 5.3: A ring R satisfying the extra condition that (R,⋅) is a monoid (with identity element denoted 1) is a unital ring, also called a ring with 1.
  • Examples:
    • The zero ring is R = {0}; denoted by 0.
    • Z,Q,R,C with + and ⋅.
    • Z/nZ with + and ⋅ modulo n,
    • Def 5.5: For any ring R and natural number n, the matrix ring M_n(R) is the set of n × n matrices with entries in R, with the standard rules for matrix addition and multiplication.
    • Def 5.6: For any abelian group G (with operation denoted by addition), the ring End_Ab(G) is the set of group endomorphisms: G → G, with operations defined by: For each f,h ∈ End_Ab(G), define f + h, f ⋅ h ∈ End_Ab(G) by (f + h)(a) = f(a) + h(a) for all a ∈ G, and f ⋅ h = f ∘ h.
    • Def 5.7: For any set X and ring R, the ring Fun(X,R) is the set of functions: X → R, with operations defined by: For each f,h ∈ Fun(X,R), define f+h, f ∘ h ∈ Fun(X,R) by (f+h)(a) = f(a) + h(a) and (f∘h)(a) = f(a) ∘ h(a)for all a ∈ X.
  • Lemma 5.9: If R is a ring, then for all a,b ∈ G:
    (1) 0a = 0 = a0.
    (2) (-a)b = -(ab) = a(-b).
    (3) (-a)(-b) = ab.
    Moreover if R is a ring with 1, then
    (4) The multiplicative identity 1 is unique.
    (5) (-1)a = -a = a(-1) for all a ∈ R.

• Section B: Homomorphisms and isomorphisms
  • Def 5.11: A (ring) homomorphism from a ring R to a ring S is a function f:R → S satisfying f(r + r') = f(r) + f(r') and f(rr') = f(r)f(r') for all r,r' ∈ R. A (ring) isomorphism from R to S is a homomorphism from R to S that is a bijection. Two rings R and S are isomorphic, written R ≅ S, if there is an isomorphism from R to S.
  • Thm 5.14: "Isomorphic" is an equivalence relation on the set of all rings. Moreover, if φ:R → S and θ:S → T are isomorphisms, then φ^-1:S → R and θ ∘ φ:R → T are also isomorphisms.
  • Def 5.15: An Isomorphism Problem algorithm determines, upon input of two rings R and S, whether or not R ≅ S. A Classification Problem algorithm enumerates (lists) all of the rings up to isomorphism.
  • Def 5.16: A (ring) isomorphism invariant is a property P of rings such that whenever R ≅ S and R has P then S has P.
  • Lemma 5.17: If P is an isomorphism invariant, R is a ring that has P, and S is a ring that does not have P, then R is not isomorphic to S.
  • Def 5.20: A ring R is commutative if for every a,b ∈ R, ab = ba.
  • Def 5.21: A ring R is a division ring if (R - {0},⋅) is a group. A ring R is a field if (R - {0},⋅) is an abelian group.
  • Example:
    • Lemma 5.22: Z/nZ is a field if and only if n is prime.
  • Def 5.24: Let R be a ring. A nonzero element a ∈ R is a zero divisor if there exists a nonzero element b ∈ R such that either ab = 0 or ba = 0.
  • Def 5.25: An integral domain is a commutative ring R with 1 ≠ 0 such that R contains no zero divisors.
  • Prop 5.26: (1)Every field is a division ring.
    (2) Every field is an integral domain.
    (3)Every division ring contains no zero divisors.
    (4) The converses of statements (1-3) are false.
  • Example:
    • Def 5.27: The (real) quaternion ring is the set H = {a + bi + cj + dk | a,b,c,d ∈ R}, with operations defined by (a + bi + cj + dk) + (a' + b'i + c'j + d'k) = (a+a') + (b+b')i + (c+c')j + (d+d')k and (a + bi + cj + dk)(a' + b'i + c'j + d'k) = (aa'-bb'-cc'-dd') + (ab'+ba'+cd'-dc')i + (ac'-bd'+ca'+db')j + (ad'+bc'-cb'+da')k.
    • Lemma 5.29: The real quaternion ring is a noncommutative division ring.
  • Def 5.30: Let R be a ring with 1 ≠ 0. A unit of R is an element r ∈ R such that there exists an s ∈ R with rs = 1 = sr. The group of units of R is the set of units of R with group operation given by the multiplication in R, and is denoted R^x.
  • Lemma 5.31: The group of units of a ring with 1 ≠ 0 is a group.
  • Examples
    • If R is a field, then R^x = R - {0}.
    • Def 5.32: If R is a commutative ring with 1, then the group of units of the ring M_n(R) is the general linear group GL_n(R).
  • Lemma 5.33: If a is a zero divisor in a ring R, then a is not a unit.
  • Prop 5.34: (1) A finite integral domain is a field. (2) A finite ring with 1 ≠ 0 that contains no zero divisors is a division ring.
  • Def 5.35: An element r of a ring R is called nilpotent if r^m = 0 for some integer m ≥ 1.
  • Thm 5.36: Let R be a ring with 1. If b is a nilpotent element of R, then 1-b is a unit of R.
  • Thm 5.37: The following are ring isomorphism invariants.
    (0) "with 1".
    (1) Group isomorphism invariants of the additive group.
    (2) Semigroup isomorphism invariants of the multiplicative semigroup.
    (3) "commutative", "division ring", "field", "integral domain".
    (4) The number of zero divisors.
    (5) Group isomorphism invariants of the group of units (including the isomorphism class).
  • More examples:
    • Def 5.38: Let R be a commutative ring with 1 ≠ 0 and let x be a letter. The polynomial ring R[x] is the set of formal expressions a₀ + a₁x + ... + a_n xⁿ with n ≥ 0 and each a_i ∈ R (called a polynomial), with operations defined by: (∑_i=1ⁿ a_i xⁱ) + (∑_j=1^q b_j x^j) = ∑_i=1^max{n,q} (a_i + b_i) xⁱ, and (∑_i=1ⁿ a_i xⁱ) ⋅ (∑_j=1^q b_j x^j) = ∑_i=1ⁿ ∑_j=1^q (a_i b_j) x^i+j.
    • Def 5.39: Let R be a commutative ring with 1 ≠ 0 and let G be a group (with operation written multiplicatively). The group ring RG is the set of formal expressions ∑_{g ∈ G} r_gg such that r_g ∈ R for all g ∈ G and r_g = 0 for all but a finite number of g's (that is, ∑_{g ∈ G} r_gg = a₁g₁ + ... + a_ng_n with n ≥ 0 and each a_i ∈ R and g_i ∈ G), with operations defined by: (∑_{g ∈ G} r_gg) + (∑_{g ∈ G} s_gg) = ∑_{g ∈ G} (r_g + s_g)g, and (∑_{g ∈ G} r_gg) ⋅ (∑_{g ∈ G} s_gg) = ∑_{g ∈ G} ∑_{h ∈ G} r_gs_h gh.
    • Def 5.40: Let R be a commutative ring with 1 ≠ 0 and let M be a monoid (with operation written multiplicatively). The monoid ring RM is the set of formal expressions ∑_{m ∈ M} r_mm such that r_m ∈ R for all m ∈ M and r_m = 0 for all but a finite number of m's, with operations defined by: (∑_{m ∈ M} r_mm) + (∑_{m ∈ M} s_mm) = ∑_{m ∈ M} (r_m + s_m)m, and (∑_{m ∈ M} r_mm) ⋅ (∑_{m ∈ M} s_mm) = ∑_{m ∈ M} ∑_{n ∈ M} r_ms_n mn.
    • Rmk: A polynomial ring is a monoid ring of the monoid {x}^* (the set of words over the alphabet {x}).
    • Def 5.41: Let R be a commutative ring with 1 ≠ 0. The noncommutative polynomial ring on n variables x₁,...,x_n with coefficients in R is the monoid ring R⟨x₁,...,x_n⟩ = R({x₁,...,x_n}^*), and the commutative polynomial ring on n variables x₁,...,x_n with coefficients in R is the monoid ring R[x₁,...,x_n] = R(Nⁿ).
    • Lemma 5.42: If R is a commutative ring with 1 ≠ 0, then: (1) R[x] and R[x₁,...,x_n] are commutative rings with 1 ≠ 0. (2) R⟨x₁,...,x_n⟩ is a ring with 1 ≠ 0, which is noncommutative if n ≥ 2. (3) If G is a group, then RG is a ring with 1 ≠ 0. (4) If M is a monoid, then RM is a ring with 1 ≠ 0.
  • The big Venn diagram of the universe of rings with 1.

• Section C: Ring and homomorphism constructions: Subrings
  • Definition and examples
    • Def 5.44: A subring of a ring R is a subset S of R that is a ring under the operations of R.
    • Lemma 5.45: A subset S of a ring R is a subring of R if and only if S is nonempty and closed under subtraction and multiplication.
    • Examples:
      • Z is a subring of Q, which is a subring of R, which in turn is a subring of C.
      • 2Z is a subring without 1 of the ring Z with 1.
      • Lemma 5.46: Let d be a squarefree integer (that is, the prime factorization of d has no repeated primes). Then the subset Q(√ d) = {a + b√ d | a,b ∈ Q} of C is a subring that is a field (called a quadratic field), and Z(√ d) = {a + b√ d | a,b ∈ Z} is a subring of Q(√ d).
      • Lemma 5.47: The set of continuous functions : [0,1] → R is a subring of Fun([0,1],R), denoted C([0,1]).
      • Lemma 5.48: If R is a ring, then the set S = {D_a | a ∈ R} of M_n(R), where D_a is the matrix with each diagonal entry equal to a and all nondiagonal entries equal to 0, is a subring of M_n(R), and R ≅ S. The subset U of M_n(R) of upper triangular matrices is also a subring of M_n(R).
    • Prop 5.49: The center of a ring R, namely the set {z ∈ R | zr = rz for all r ∈ R}, is a commutative subring of R.
    • Prop 5.50: (1) If R is a commutative ring with 1 ≠ 0, then: (1a) R is a subring of R[x]. (1b) R is a subring of R[x₁,...,x_n]. (1c) R is a subring of R⟨x₁,...,x_n⟩.
      (2) If S is a subring containing 1 of a commutative ring R with 1 ≠ 0, then (2a) S[x] is a subring of R[x]. (2b) S[x₁,...,x_n] is a subring of R[x₁,...,x_n]. (2c) S⟨x₁,...,x_n⟩ is a subring of R⟨x₁,...,x_n⟩.
      (3a) If R is a commutative ring with 1 ≠ 0 and H is a subgroup of a group G, then RH is a subring of RG.
      (3b) If R is a commutative ring with 1 ≠ 0 and M' is a submonoid of a monoid M, then RM' is a subring of RM.
  • Interactions with ring homomorphisms
    • Thm 5.52: Restrictions and extensions of ring homomorphisms are ring homomorphisms. In particular, inclusions are injective ring homomorphisms.
    • Thm 5.53: Homomorphic images and preimages of subrings are subrings.
    • Def 5.54: Let f: R → S be a ring homomorphism. The kernel of f is the subring Ker(f) = f^-1({0}) = {r ∈ R | f(r) = 0} of R. The image of f is the subring Im(f) = f(R) of S.
    • Def 5.55: An embedding of a ring R in a ring S is an injective homomorphism :R → S.
    • Examples:
      • Embedding M₂(R) in M₃(R).
  • Interactions with isomorphism invariants
    • Prop 5.57: Every subring of a commutative ring is commutative.
    • Prop 5.58: Every subring with 1 ≠ 0 of an integral domain is an integral domain.
    • Prop 5.59: There exists a subring with 1 ≠ 0 of a field that is not a field. There exists a subring without 1 of a field.

• Section D: Ring and homomorphism constructions: Product rings
  • Definition and examples
    • Def 5.61: (a) Let R_i be a ring for all 1 ≤ i ≤ n. The direct product of the rings R_i is the Cartesian product R₁ × R₂ × ... × R_n with addition defined by (r₁,...,r_n)(s₁,...,s_n) = (r₁+s₁,...,r_n+s_n) and multiplication defined by (r₁,...,r_n)(s₁,...,s_n) = (r₁s₁,...,r_ns_n).
      (b) Let R_α be a ring for all α in an index set J. The direct product of the rings R_α is the Cartesian product Π_{[α ∈ J]} R_α with addition defined by (r_α)_{α ∈ J}(s_α)_{α ∈ J} = (r_α+s_α)_{α ∈ J} and multiplication defined by (r_α)_{α ∈ J}(s_α)_{α ∈ J} = (r_αs_α)_{α ∈ J}. The direct sum of the rings R_α is the subset ⊕_{[α ∈ J]} R_α of the direct product Π_{[α ∈ J]} R_α given by ⊕_{[α ∈ J]} R_α = {(r_α)_{α ∈ J} | r_α = 0_α for all but finitely many α}, with the same multiplication as the direct product.
    • Thm 5.62: The direct product of a collection of rings is a ring, and the direct sum of the collection is a subring of the direct product.
    • Examples
  • Interactions with ring homomorphisms
    • Thm 5.64: (1) Projections are ring homomorphisms. (2) Inclusions into products are embeddings. (3) Products of ring homomorphisms (with the same domain; see Def 2.57) are ring homomorphisms.
    • Thm 5.65: If R_α ≅ S_α for all α ∈ J, then Π_{[α ∈ J]} R_α ≅ Π_{[α ∈ J]} S_α. (2) R × S ≅ S × R.
  • Interactions with isomorphism invariants and ring constructions
    • Thm 5.67: (1) A direct product Π_{[α ∈ J]} R_α of rings R_α is a ring with 1 if and only if R_α is a ring with 1 for all α ∈ J. (2) A direct product Π_{[α ∈ J]} R_α of rings R_α is commutative if and only if R_α is commutative for all α ∈ J.
    • Thm 5.68: Every direct product of two or more nonzero rings contains zero divisors.
    • Thm 5.69: If S_α is a subring of R_α for all α ∈ J, then Π_{[α ∈ J]} S_α is a subring of Π_{[α ∈ J]} R_α.
    • Examples

• Section E: Ring and homomorphism constructions: Quotient rings
  • Definition and examples
    • Def 5.74: An equivalence relation ~ on a ring R is compatible with addition and multiplication if whenever r,s,t ∈ R and r ~ s then r+t ~ s+t, rt ~ st, and tr ~ ts.
    • Rmk 5.75: Let R be a ring and let ~ be an equivalence relation on R that is compatible with addition and multiplication. The set R/~ of equivalence classes, with addition [r]+[s] = [r+s] and multiplication [r][s] = [rs], is a ring if and only if ~ is compatible with addition and multiplication.
    • Def 5.76: An ideal (or 2-sided ideal) of a ring R is an additive subgroup I of R satisfying the property that rI ⊆ I and Ir ⊆ I for all r ∈ R.
    • Lemma 5.77: A subset I of a ring R is an ideal of R if and only if I is nonempty, closed under subtraction, and closed under multiplication (on either side) by elements of R.
    • Prop 5.78: Let I be a subring of a ring R. The following are equivalent: (1) I is an ideal of R. (2) The equivalence relation ~_I induced by the left action of the additive group I on the additive group R is compatible with addition and multiplication. (3) The quotient group R/I (under addition) is a ring with multiplication (r+I)(s+I) = rs+I.
    • Def 5.80: Let I be an ideal of a ring R. The quotient ring is the set R/I of left cosets of the additive group I on the additive group R, with the operations (r+I) + (s+I) = (r+s)+I and (r+I)(s+I) = rs+I for all r+I,s+I ∈ R/I.
    • Cor 5.81: Let I be an ideal of a ring R. The quotient ring R/I is a ring.
    • Examples
      • For any ring R, {0},R are ideals of R, and R/{0} ≅ R and R/R ≅ 0.
      • For any ring R with 1 and ideal I, 1 ∈ I if and only if I = R.
      • 2Z is an ideal of Z and Z/2Z = Z/2Z.
      • For any commutative ring R with 1, the subset xR[x] of R[x] is an ideal and R[x]/xR[x] ≅ R.
  • Interactions with homomorphisms
    • Thm 5.84: Let I be an ideal of a ring R. The quotient map q:R → R/I defined by q(r) = r+I is a ring homomorphism with kernel Ker(q) = I.
    • Cor 5.85: A subring I of a ring R is an ideal in R if and only if I is the kernel of a ring homomorphism with domain R.
    • Prop 5.88: (Homomorphism Building Theorem (HBT) for Quotient Rings): Let I be an ideal of a ring R and let f: R → S be a ring homomorphism satisfying the property that I ⊆ Ker(f). Then:
      (1) There is a well-defined homomorphism φ:R/I → S defined by φ(r+I) = f(r) for all r+I in R/I.
      (2) If f is onto, then φ is onto.
      (3) If f also satisfies the property that Ker(f) ⊆ I, then φ is one-to-one.
    • Cor 5.89: (First Isomorphism Theorem for Rings(1ITR)): If f: R → S is a ring homomorphism, then Ker(f) is an ideal of R, f(R) is a subring of S, and R/Ker(f) ≅ f(R).
    • Cor 5.90: If f:R → S is a ring homomorphism, then f is injective if and only if Ker(f) = {0}.
    • Examples
      • Lemma 5.92: For any commutative ring R with 1, the kernel of the ring homomorphism f: R[x] → R, defined by f(p) = p(0) for all p ∈ R[x], is the ideal xR[x].
      • Lemma 5.93: For any ideal I of a commutative ring R with 1, I[x] is an ideal of R[x] and R[x]/I[x] ≅ (R/I)[x].
      • Lemma 5.94: For any ideal I of a ring R, M_n(I) is an ideal of M_n(R), and M_n(R)/M_n(I) ≅ M_n(R/I).
      • For any commutative ring with 1 ≠ 0 and any group G with normal subgroup N, the group ring RN is an ideal of RG if and only if N = G.
      • R[x]/((x² - 1)(R[x])) ≅ C.
  • Interactions with isomorphism invariants
    • Thm 5.97: (1) Every quotient of a ring with 1 is a ring with 1.
      (2) Every quotient of a commutative ring is a commutative ring.
    • Rmk 5.98: There is an example of a quotient of a field that is not an integral domain.
  • Interactions with ring constructions
    • Prop 5.100: Let R be a ring. Let I,J be ideals of R. Then I+J, I ∩ J, and IJ = {∑_k=1ⁿ i_kj_k | n ≥ 0 and each i_k ∈ I and j_k ∈ J} are ideals of R.
    • Thm 5.101: (Second Isomorphism Theorem for Rings (2ITR)): Let S be a subring and let I be an ideal of R. Then S + I = {s + i | s ∈ S, i ∈ I} is a subring of R, S ∩ I is an ideal of S, and (S+I)/I ≅ S/(S ∩ I).
    • Thm 5.102: (Third Isomorphism Theorem for rings (3ITR)): Let I and J be ideals of a ring R with I ⊆ J. Then J/I is an ideal of R/I and (R/I)/(J/I) ≅ R/J.
    • Prop 5.104: Homomorphic images (sort of) and preimages of ideals are ideals. That is: Let R and S be rings and let f: R → S be a ring homomorphism.
      (1) If I is an ideal of R then f(I) is an ideal of f(R).
      (2) If J is an ideal of S then f^-1(J) is an ideal of R.
    • Thm 5.105: (Lattice Isomorphism Theorem for Rings (LITR)): Let I be an ideal of a ring R, and let q: R → R/I be the quotient map. Then the function Ψ : {subrings of R containing I} → {subrings of R/I} defined by Ψ(S) = q(S) = S + I is a bijection with inverse defined by Ψ^-1(T) = q^-1(T) for each subring T of R/I. Moreover, for a subring S of R containing I, S is an ideal of R if and only if S/I is an ideal of R/I (that is, Ψ and Ψ^-1 preserve ideals).

• Section F: Presenting rings, and ring homomorphism building theorems
  • Prop 5.108: (Homomorphism Building Theorem (HBT) for Z and Z/nZ): Let S be a ring with 1. (a) If R = Z, then for there is a unique ring homomorphism f: R → S such that f(1_R) = 1_S. (b) If R = Z/nZ and n ⋅ 1_S = 0, then there is a unique ring homomorphism f: R → S such that f(1_R) = 1_S.
  • Rmk: The "free", or largest, ring with 1 ≠ 0 that can be "generated" by a single element x is Z[x].
  • Prop 5.109: (Homomorphism Building Theorem (HBT) for Polynomial Rings): Let R and S be commutative rings, let h:R → S be a ring homomorphism, and let j: {x₁,...,x_n} → S be a function. Then:
    (1) There is a unique ring homomorphism f: R[x₁,...,x_n] → S such that f|_R = h and f(x_i) = j(x_i) for all i (namely f(∑_{[e1,...,en ≥ 0]} r_e1,...,en x₁^e₁ ... x_n^e_n) = ∑_{[e1,...,en ≥ 0]} h(r_e1,...,en) (j(x₁))^e₁ ... (j(x_n))^e_n).
    (2) There is a unique ring homomorphism f: R⟨x₁,...,x_n⟩ → S such that f|_R = h and f(x_i) = j(x_i) for all i.
  • Cor 5.110: If R and S are commutative rings and h:R → S is a ring homomorphism, then there is a unique ring homomorphism f: R[x₁,...,x_n] → S[x₁,...,x_n] such that f|_R^S = h and f(x_i) = h(x_i) for all i (namely f(∑_{[e1,...,en ≥ 0]} r_e1,...,en x₁^e₁ ... x_n^e_n) = ∑_{[e1,...,en ≥ 0]} h(r_e1,...,en) x₁^e₁ ... x_n^e_n).
  • Def 5.113: Let R be a ring and let A ⊆ R.
    (1) The subring of R generated by A is the set of all sums and differences of products of elements of A; that is, {(∑_i=1^m a_i1...a_ini) - (∑_i=1^m' a_i1'...a_ini'')| m,m' ≥ 0, each n_i,n_i' ≥ 1, and each a_ij,a_ij' ∈ A}.
    (2) The ideal of R generated by A, denoted (A), is the set of all sums and differences of elements of A and elements of A multiplied on the left and/or right by elements of R. If A = {a}, then (A) is denoted (a).
  • Prop 5.114: Let R be a ring and let A ⊆ R. Then:
    (0) If R is a ring with 1 ≠ 0, then (A) = {∑_i=1^m r_ia_is_i | m ≥ 0, each a_i ∈ A, and each r_i,s_i ∈ R}.
    (1) (A) is an ideal.
    (2) (A) is the smallest ideal of R containing A.
    (3) (A) = ∩_{[A ⊆ I and I is an ideal of R]} I; that is, (A) is the intersection of all ideals of R containing A.
  • Prop 5.115: Let R be a ring and let A ⊆ R. If S is the subring of R generated by A, then S is the smallest subring of R containing A, and S is the intersection of all subrings of R containing A.
  • Def 5.116: Let I be an ideal of a ring R. The ideal I is principal if I = (a) for some a ∈ R, and I is finitely generated if I = (A) for some finite subset A of R.
  • Examples
    • Prop 5.118: For any ring R with 1, the subring of R generated by 1 is (ring) isomorphic to either Z or Z/nZ for some n.
    • Def 5.119: Let R be a ring with 1. If the subring of R generated by 1 is isomorphic to Z/nZ, then n is called the characteristic of R; if the subring generated by 1 is isomorphic to Z then the characteristic of R is 0.
    • Lemma 5.120: The characteristic of a ring with 1 is an isomorphism invariant.
    • For any ring R with 1, (1) = R.
    • Z[x] is generated as a ring by {1,x}. Z[x] is generated as a group under addition by {xⁿ | n ≥ 0} but not by {1,x}.
    • Let J be the ideal of R = Z[x] defined by J = (2,x²+x+1) and let I be the ideal of S = Z/2Z[x] defined by I = (x²+x+1). Then R/J ≅ S/I is a field, and the only ideals of R containing J are J and R.
    • Prop 5.121: A commutative ring R with 1 ≠ 0 is a field if and only if the only ideals of R are 0 and R.
    • 2Z ≇ 3Z.
  • Thm 5.123: (Homomorphism Building Theorem (HBT) for Quotients of Polynomial Rings): Let R be a commutative ring with 1 ≠ 0, let S be a commutative ring, let h:R → S be a ring homomorphism, let A ⊆ R[x₁,...,x_n], and let j: {x₁,...,x_n} → S be a function satisfying the property that whenever ∑_i=1^m r_i x₁^e_i,1 ... x_n^e_i,n ∈ A (with m ≥ 1 and each r_i ∈ R and e_i,j ≥ 0) then ∑_i=1^m h(r_i) (j(x₁))^e_i,1 ... (j(x_n))^e_i,n = 0. Then there is a unique ring homomorphism f: R[x₁,...,x_n] / (A) → S such that f(r+(A)) = h(r) for all r ∈ R and f(x_i+(A)) = j(x_i) for all i.
  • More examples
  • Thm 5.124: (Homomorphism Building Theorem (HBT) for Group Rings): Let R be a commutative ring with 1 ≠ 0, let G and H be groups with identity elements e_G and e_H respectively, and let h: G → H be a group homomorphism. Then there is a unique ring homomorphism f: RG → RH satisfying f(g) = h(g) for all g ∈ G and f(re_G) = re_H for all r ∈ R.