Reminder: The "How to" guide for proof writing, including proper referencing of prior results, should be followed when writing up your solutions.

• Problem Set 1:
  • E0.1: Let f:G → H be a function and let C,D be subsets of G and let E,F be subsets of H. Prove the following parts of TOC Thms 0.7-0.9:
    (a) f(C ∪ D) = f(C) ∪ f(D).
    (b) f ^-1(E ∪ F) = f ^-1(E) ∪ f ^-1(F).
    (c) f(f ^-1(E)) ⊆ E.
    (d) Find an example of sets G and H, a function f:G → H, and a subset E of H, for which f(f ^-1(E)) ≠ E.
  • E0.2: For each of the following examples of a set X and an equivalence relation ~ on X, find and prove a bijection between the set X/~ of equivalence classes and a subset of X using TOC Thm 0.17.
    (a) X = R². Define (x₀,y₀) ~ (x₁,y₁) if and only if x₀ - x₁ and y₀ - y₁ are both even integers.
    (b) X is the square ([-1,1] × {-1,1}) ∪ ({-1,1} × [-1,1]). Define (x₀,y₀) ~ (x₁,y₁) if and only if either (there is an ε in {1,-1} such that x₁ = ε x₀ and y₁ = ε y₀) or (there is an ε in {1,-1} such that x₁ = ε y₀ and y₁ = - ε x₀).
  • E1.1: Text exercise 1.1.6.
  • E1.2: Let M be the set of all "words" that can be written in the letters a,b,f,g. (That is, the words a,b,afg,abgfafffab are elements of M, as is the word with no letters (called the empty word and written as λ).) Let T be the intersection of all of the equivalence relations on M that satisfy the properties that (i) ab ~ ba ~ λ ~ fg ~ gf, and (ii) whenever u,v,w,x ∈ M and v ~ w then uvx ~ uwx.
    (a) Show that T is the smallest equivalence relation on M satisfying (i-ii). (That is, show that the subset T of M × M is an equivalence relation on M satisfying (i-ii), and show that if T' is another equivalence relation satisfying (i-ii), then T ⊆ T'. Notation: Use ~_T to denote the relation from T, and ~_T' to denote the relation from T', etc.)
    (b) Define a binary operation on the set M/~_T of equivalence classes by [u][v]=[uv]. Show that M/~_T is a group.
  • E1.3: Let U(n) (or (Z/nZ)^x) be the set of all positive integers less than n that are relatively prime to n, with the binary operation of multiplication modulo n; this is a group (you don't need to prove that). List the elements of U(18), find the identity of U(18), and find the inverse of each element of U(18).
  • E1.4: A symmetry of a subset X of the Euclidean plane R² 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.) For each of the following subsets X of the plane R², describe all of the elements of the group of symmetries. (Briefly explain how you got your answer; full proof not needed.)
    (a) An infinitely long strip of equally spaced R's: ... R R R R R R R R ...
    (b) An infinitely long strip of equally spaced S's: ... S S S S S S S S ...
    (c) An infinitely long strip of equally spaced B's: ... B B B B B B B B ...
    (d) An infinitely long strip of equally spaced H's: ... H H H H H H H H ...
  Due 9/1/17 for grading: E0.1(c-d), E1.2, E1.4(a-b)

• Problem Set 2:
  • E1.5: Let G be a group with elements a and b. Show that the following are equivalent: (i) ab = ba. (ii) aba^-1b^-1 = e. (iii) For all integers j and k, a^jb^k = b^ka^j. (iv) For all integers n, (ab)ⁿ = aⁿbⁿ.
  • E1.6: Let g be an element of a group G.
    (a) Prove that |g|=|g^-1|.
    (b-i) Prove that if |g| = n for a natural number n, then e,g,...,g^n-1 are all distinct elements of G.
    (b-ii) Prove that if |g| = ∞, then for all integers i,j with i ≠ j, gⁱ ≠ g^j.
    (b-iii) Prove (using (b-i-ii)) that |g| ≤ |G|.
    (c) Prove that if f:G → H is an isomorphism, then |g| = |f(g)|.
    (d) Give an example of a group G and elements g and h satisfying |g| is finite, |h| is finite, and |gh| is infinite.
  • E1.7: Show that if G and H are groups, φ,σ: G → H are homomorphisms, S ⊆ G is a generating set for G (that is, G = ⟨ S ⟩), and φ(t) = σ(t) for all t ∈ S, then φ = σ (that is, φ(g) = σ(g) for all g ∈ G).
  • E1.8: In each part, determine (and prove) whether or not the group G is isomorphic to the group H.
    (a) G = U(20) and H = U(24).
    (b) G = Aut(Z/4Z) and H = Z/3Z.
    (c) G = GL₂(Z/2Z) and H = D₆.
    (d) G = Q₈ and H = S₄.
    (e) G = Z and H = Q.
  • E1.9: Let n be a natural number and let σ be an element of the symmetric group S_n.
    (a) Show that if σ is an m-cycle (a₁ a₂ ... a_m), then |σ| = m.
    (b) Show that if σ is a product of disjoint cycles of orders m₁,...,m_k, then |σ| is the least common multiple of the m_i's (that is, |σ| = lcm(m₁,...,m_k)). (Note: You may use the results in E1.5 in this problem.)
  • E1.10: Show that if G and H are groups and G ≅ H, then Aut(G) ≅ Aut(H).
  Due 9/8/17 for grading: E1.6(c,d), E1.8(a,b), E1.9(a), E1.10

• Problem Set 3:
  • E1.11: (a) Prove TOC Lemma 1.49. (b) Prove TOC Lemma 1.51.
  • E1.12: For each of the following groups G and generating sets A, draw the Cayley graph Γ(G,A).
    (a) G = D₆, A = {s,r}
    (b) G = D₆, A = {s,rs}
    (c) G = S₃, A = {(12),(13)}
  • E2.1: (a) Prove TOC Thm 2.3. (b) Prove TOC Thm 2.15.
  • E2.2: 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).)
    (a) For the following groups G and subsets H, determine the elements of the sets C_G(H) and N_G(H) (and prove your answer).
    (a-i) G = D_2n, H={e,s} where s is reflection of the regular n-gon around a line through a vertex and the center point of the n-gon.
    (a-ii) G = SL₂(Z/3Z), H = {p_n | n ∈ Z/3Z} where p_n is the matrix with top row [1 n] and bottom row [0 1].
    (a-iii) G = F({a,b}), H = {a}
    (a-iv) G = Q (quaternions), H = {1,-1,i,-i}
    (b-i) Show that if a ∈ G, then C_G({a^i | i ∈ Z}) = C_G(a). Must N_G({a^i | i ∈ Z}) = N_G(a) also hold?
    (b-ii) Show that if A ⊆ G, then C_G(⟨ A ⟩) = C_G(A). Must N_G(⟨ A ⟩) = N_G(A) also hold?
    (c) Prove that C_G(H) is a subgroup of N_G(H), and N_G(H) is a subgroup of G.
  • E2.3: Prove the Homomorphism Building Theorem 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, there is a unique group homomorphism f: G → H such that f(a) = y.
  Due 9/15/17 for grading: E1.11(a)(2), E1.12(b), E2.1(a)(2), E2.2(a-iv),(b-i)

• Problem Set 4:
  • E2.4: (a) Prove that for any group G, Z(G) = C_G(G) and G = C_G(1).
    (b) Let H be the subgroup of SL₃(Z) of all matrices with top row [1 a b], middle row [0 1 c], and bottom row [0 0 1] (where a,b,c ∈ Z) (with group operation given by matrix multiplication); this is called the Heisenberg group. Determine the elements of Z(H).
    (c) Show that the order of the center of a group is an isomorphism invariant.
  • E2.5: For each natural number n, let V_n be the subset of the symmetric group S_n defined by
    V_n = {(i j)(k m) | i,j,k,m ∈ {1,...,n}, i ≠ j, and k ≠ m};
    that is, V_n is the set of all products of two 2-cycles. Let A_n be the subgroup of S_n generated by V_n; the group A_n is called the alternating group of degree n. For n = 3, n = 4, and n = 5 determine whether or not A_n = V_n, and whether or not A_n = S_n.
  • E2.6: Let G, H, and K be groups.
    (a) For each g ∈ G, let σ_g,G:G → G be the automorphism of G defined by σ_g,G(x) = gxg^-1 for all x ∈ G. (If there is only one group in the discussion, this automorphism is also written σ_g.) Show that the subset Inn(G) = {σ_g,G | g ∈ G} of Aut(G), with the same group operation, is a subgroup of Aut(G); this subgroup is called the inner automorphism group of G.
    (b) Find (and prove) an example of a subgroup K in a group G for which the map Inn(K) → Inn(G) given by σ_k,K → σ_k,G for all k ∈ K is not a well-defined function.
    (c) Show that Aut(G) × Aut(H) embeds in Aut(G × H).
    (d) Find (and prove) an example of groups G and H for which Aut(G) × Aut(H) is not isomorphic to Aut(G × H).
  • E2.7: Show that if f_i:G_i → H_i is a group homomorphism for i=1,2, then the function f':G₁ × G₂ → H₁ × H₂ defined by f'(g₁,g₂) = (f₁(g₁),f₂(g₂)) can be constructed from the functions f_i and projections using products and compositions of functions. Use your construction and various TOC theorems to show that f' is a homomorphism.
  • E2.8: (a) Prove TOC Thm 2.59. (b) Prove TOC Thm 2.66.
  • E2.9: In each part, determine whether the group G is isomorphic to the group H (and prove your answer).
    (a) G = C₈ × C₂ and H = C₄ ⊕ C₄.
    (b) G = S₃ × U(3) and H = D₁₂.
    (c) G is the subgroup of SL₃(Z/3Z) of all matrices with top row [1 a b], middle row [0 1 0], and bottom row [0 0 1] (where a,b ∈ Z/3Z) (with group operation given by matrix multiplication), and H = C₃ × C₃.
    (d) G = Z and H = Z × Z.
    (e) G = Z³ = Z × Z × Z, and H is the Heisenberg group in E2.4(b).
  Due 9/22/17 for grading: E2.4(b,c), E2.6(b,c,d), E2.9(e)

• Problem Set 5:
  • E3.1: Let G be a group, let N be a normal subgroup of G, and let G/N be the quotient group. Prove the following.
    (a) If G is finite, then G/N is finite and |G/N| divides |G|.
    (b) If g is an element of G and N ⊴ G, then |g| ≥ |gN|. Moreover, if g has finite order in G, then gN has finite order in G/N and |gN| divides |g|.
    (c) If G is abelian then G/N is abelian.
    (d) If G is cyclic then G/N is cyclic.
    (e) If G is finitely generated then G/N is finitely generated.
    (Note: These are TOC Theorems 3.50, 3.52, 3.55, 3.60, and 3.63.)
  • E3.2: Let G be a group.
    (a) Let H be a group. Show that {1_G} × H ⊴ G × H, and (G × H)/({1_G} × H) ≅ G.
    (b) Let M and N be normal subgroups of G satisfying G = MN. Prove that M ∩ N ⊴ G, and that G/(M ∩ N) ≅ (G/M) × (G/N).
  • E3.3: Let G be a group. For each g,h ∈ G let [g,h] denote the element [g,h] = ghg^-1h^-1 of G. ([g,h] is the commutator of g with h.) Let G' = ⟨ {[g,h] | g,h ∈ G} ⟩, the commutator subgroup of G.
    (a) Prove that G' ⊴ G.
    (b) Let N ⊴ G. Prove that G/N is abelian if and only if G' ⊆ N.
    (Note: This problem is asking you to prove TOC Prop 3.58.)
  • E3.4: Give an example of a group G and a normal subgroup N ≠ {1} such that G ≅ G/N.
  • E3.5: Let G be a group, let H ≤ G, and let N ⊴ G such that NH = G and N ∩ H = {1}.
    (a) Let N × H be the Cartesian product set, and let f:N × H → G be the function defined by f(n,h) = nh for all n ∈ N and h ∈ H. Show that f is a bijection.
    (b) Find a group operation on the set N × H that makes f an isomorphism. (Note: The group operation will not be the operation for the direct product, in general. Given (n,h),(n',h') ∈ N × H, you need to express the product (n,h)(n',h') in the form (n'',h''); that is, you need to write n'' and h'' in terms of n,n',h,h'.)
    (c) Show that the group operation you found in (b) is the same as the direct product multiplication if and only if H ⊴ G.
  • E3.6: Let G = ⟨ A | R ⟩.
    (a) Suppose that s is an element of the normal subgroup ⟨ R ⟩^N of F(A). Show that G ≅ J where J = ⟨ A | R ∪ {s} ⟩.
    (b) Suppose that b is a letter not in A, and w is an element of F(A). Show that G ≅ K where K = ⟨ A ∪ {b} | R ∪ {b=w} ⟩.
    (c) The infinite dihedral group D_∞ is presented by D_∞ = ⟨ r,s | s² = 1, sr = r^-1s ⟩. Use the operations for changing presentations for isomorphic groups in parts (a) and (b) (these are called Tietze transformations) along with TOC Thm 1.16 to show that D_∞ ≅ L where L = ⟨ a,b | a² = 1, b² = 1 ⟩.
    (Hint/note: In parts (a) and (b), use the HBT for presentations, TOC Thm 3.45.)
  Due 9/29/17 for grading: 3.1(b,d), 3.3, 3.5(b), 3.6(a)

• Problem Set 6:
  • E3.7: Let G be the group with presentation
    G = ⟨ a,b,c | a²=1, b²=1, c²=1, (ab)³=1, (ac)²=1, (bc)³=1 ⟩.
    (a) Prove that G ≅ S₄.
    (b) Show that there is a surjective homomophism f:G → Z/2Z such that f(a)=f(b)=f(c)=1.
    (c) Let h:S₄ → Z/2Z be the composition f ∘ j where j is the isomorphism from S₄ to G from part (a). What group is the kernel of h? (Only the name of the group is required; proof not needed.)
    [(d) Challenge: Show that for all n ≥ 3,
    S_n ≅ ⟨ a₁,...,a_n-1 | a_i² = 1 for all 1 ≤ i ≤ n-1, (a_ia_i+1)³ = 1 for all 1 ≤ i ≤ n-2, (a_ia_j)² = 1 for all 1 ≤ i < i+2 ≤ j ≤ n-1 ⟩. Use this to show that there is a surjective homomophism f:S_n → Z/2Z such that f(a_i)=1 for all 1 ≤ i ≤ n-1. Determine the kernel of f.]
  • E3.8: Let G be a group, and let H be a subgroup of G of finite index |G:H| = n.
    (a) Show that if n = 2, then H is normal in G.
    (b) Show that if n = p is a prime number, H is a normal subgroup of G, and K is another subgroup of G, then either K ≤ H or G = KH.
    (c) Show that if K ≤ H ≤ G and the index |H:K| = m of K in H is finite, then the index of K in G is finite and |G:K| = mn. (Note: You may not assume that G is a finite group.)
    (d) Show that there exists a normal subgroup N of G such that N ≤ H and |G:N| ≤ n!. (Tip: Define the function f:G → Perm(G/H) (where G/H is the set of left cosets of H in G) by (f(g))(g'H) = (gg')H for all g ∈ G and g'H ∈ G/H. Show that f is a group homomorphism, and consider the kernel and image of f.)
  • E3.9: Show that if A ⊴ G and B ⊴ H, then A × B ⊴ G × H and (G × H)/(A × B) ≅ (G/A) × (H/B).
  • E3.10: Let P be an isomorphism invariant property. A group G has the property poly-P if there is a sequence of groups
    {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 and the quotient group N_i+1/N_i has the property P.
    (a) Show that every group with the property P also has the property poly-P.
    (b) For the following properties P, decide whether every poly-P group also has the property P (and prove your answer):
    (b-i) Finite.
    (b-ii) Abelian.
    (b-iii) Cyclic.
    (c) Show that every subgroup H of a poly-abelian group G is poly-abelian. (Tip: Intersect the groups N_i in the sequence for G with the group H.)
  • E3.11: A group G is simple if |G| > 1 and the only normal subgroups of G are {e} and G. Show that if G is abelian and simple, then G ≅ Z/pZ for some prime number p.
  Due 10/6/17 for grading: E3.7(a,b,c), E3.8(b,d), E3.10(c)

• Exam 1 Monday October 9 6:00-8:00pm

• Problem Set 7:
  • E4.1: Let G be a group and let g be an element of G.
    (a) Suppose that G acts on a set A, and a ∈ A. Show that g Stab_G(a) g^-1 = Stab_G(g · a).
    (b) Show that if S is a subset of G, then g C_G(S) g^-1 = C_G(g S g^-1).
  • E4.2: Suppose that π : G → Perm(G) is the permutation representation induced by the conjugation action of G on G.
    (a) Show that π is faithful if and only if Z(G) is trivial.
    (b) If π is transitive, what can you conlude about the group G?
  • E4.3: (a) Number the elements e,(1~2),(1~3),(2~3),(1~2~3),(1~3~2) of G = S₃ by 1,2,3,4,5,6, respectively. Let π : G → Perm(G) = S₆ be the permutation representation induced by the left multiplication action of G on G. Write out the disjoint cycle notation for the image under π of each of the elements of S₃.
    (b) Let g ∈ A_n, let [g]_An denote the conjugacy class of g in A_n, and let [g]_Sn denote the conjugacy class of g in S_n. Show that either [g]_Sn = [g]_An or [g]_Sn is the union of two conjugacy classes in A_n of the same size.
  • E4.4: Let G be a group.
    (a) Show that if G/Z(G) is cyclic, then G is abelian.
    (b) Show that if |G| is a prime number, then G is cyclic.
    (c) Suppose that |G|=15 and G is not abelian. Show that Z(G) = 1 and that G has exactly 3 conjugacy classes of size 3.
    (Note: You may not use TOC Cor 4.54 in this problem (since this problem is part of its proof in the special case that p = 3 and q = 5).)
  • E4.5: A be a set and let G = Perm(A); that is, G acts on A by permutations. The fixed set of a subset S ⊆ G is Fix(S) = {a ∈ A | t(a) = a for all t ∈ S}. Show that if H ≤ G and g ∈ N_G(H), then g ∈ SetStab(Fix(H)).
  • E4.6: Let g₁,...,g_k be (unique) representatives of (all of) the conjugacy classes of a finite group G, and suppose that these elements commute pairwise.
    (a) Show that for all i ∈ {1,...,k}, |G:C_G(g_i)| ≤ |G|/k.
    (b) Show that G is abelian. (Hint: Use part (a) and the class equation.)
  Due 10/20/17 for grading: 4.2(b), 4.5, 4.6

• Problem Set 8:
  • E4.7: For the groups G = Z/5Z and H = S₃, compute the groups Aut(G × H), Inn(G × H), and Out(G × H).
  • E4.8: 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). Show that N ⋊_φ K ≅ N ⋊_θ K.
    (Hint: Show that there is an integer m such that σ φ(x) σ^-1 = θ(x)^m for all x ∈ K, and that the map ψ: N ⋊_φ K → N ⋊_θ K defined by ψ(n,x) = (σ(n),x^m) is a homomorphism.)
  • E4.9: Let N = ⟨ x | x^m = 1 ⟩ and K = ⟨ y | yⁿ = 1 ⟩ be cyclic groups of orders m and n, respectively. Let j be a natural number satisfying gcd(j,m) = 1 and jⁿ = 1 (mod m).
    (a) Show that there is a unique homomorphism φ: K → Aut(N) satisfying φ(y)(x) = x^j.
    (b) If φ is the homomorphism from part (a), show that N ⋊_φ K ≅ ⟨ r,s | r^m = 1, sⁿ = 1, srs^-1 = r^j ⟩.
    (Note: You may not use TOC Cor 4.45 or TOC Thm 4.48 in this problem (since this problem is a proof of Cor 4.45 and of a special case of Thm 4.48).)
  • E4.10: Let M be a normal subgroup of a group G, and let q: G → G/M be the quotient map; then G is an extension of M by G/M. The group G is called a split extension of M by G/M if there is a homomorphism s: G/M → G satisfying q ∘ s = Id_G/M.
    (a) Show that if N and K are groups and φ: K → Aut(N) is a homomorphism, then the semidirect product G = N ⋊_φ K is a split extension of N' = {(n,1) | n ∈ N} by G/N'.
    (b) Show that if N ⊴ G and G is a split extension of N by G/N, then G ≅ N ⋊_φ G/N for some homomorphism φ.
    (c) Give an example of a group G and a normal subgroup N of G such that G is not a split extension of N by G/N.
    (Hint: Part (b) of this problem is quite similar to E3.5.)
    (Note: You may not use TOC Thms 4.41, 4.47, 4.49 in this problem since this problem is part of their proof.)
  Due 10/30/17 at 9:30am for grading: E4.9, E4.10(a)

• Problem Set 9:
  • E4.11: (a) Prove TOC Lemma 4.52. (b) Prove TOC Cor 4.56.
  • E4.12: Prove TOC Cor 4.54(2) and (4). You may use (without proof) the following fact: If q is a prime number, then (Z/qZ)^x is a cyclic group. [We'll prove this in Math 818; it's proved in the text in Prop 18 page 314 for those of you who want to check it out now.]
  • E4.13: Prove that the quaternions are not isomorphic to a semidirect product of two nontrivial groups.
  • E4.14: Compute the Sylow subgroups of the following groups. (a) D₁₂. (b) S₃ × Z/3Z. (c) S₄.
  • E4.15: Let G be a finite group, let N ⊴ G, let p be a prime number, and let Q be a Sylow p-subgroup of G.
    (a) Show that Q ∩ N ∈ Syl_p(N).
    (b) Show that QN/N ∈ Syl_p(G/N).
    (c) Show that |Syl_p(G/N)| ≤ |Syl_p(G)|.
  • E4.16: Let G be a finite group of order pqr with p ≤ q ≤ r prime numbers (that are not necessarily distinct).
    (a) If p < q < r, prove that G has a normal Sylow subgroup (for at least one of p,q,r).
    (b) If p = q < r, show that G is not simple.
    (c) If p < q = r, show that G is not simple.
    (d) Prove that G is solvable.
    (Hint for (a) and (b): Use step (3) in the flow chart in TOC Rmk 4.75.)
  • E4.17 (a) For each N ∈ {C₇ = ⟨ a ⟩, C₄ = ⟨ b ⟩, C₂ × C₂ = ⟨ c ⟩ × ⟨ d ⟩}, list the elements of Aut(N), and determine the order of each element of Aut(N).
    (b) List all of the group homomorphisms C₇ → Aut(C₄), C₇ → Aut(C₂ × C₂), C₄ → Aut(C₇), and C₂ × C₂ → Aut(C₇).
    (c) Classify all of the groups of order 28, up to isomorphism. (Hint: Use the flow chart in TOC Rmk 4.75.)
  Due 11/8/17 at 11:30am for grading: E4.11(b), E4.15(a), E4.17(c, plus any parts of a and b that are relevant to part c)

• Problem Set 10:
  • E4.18: Let K and N be groups and let φ,θ: K → Aut(N) be homomorphisms. Show that if there is an automorphism β ∈ Aut(K) such that θ = φ ∘ β, then N ⋊_φ K ≅ N ⋊_θ K.
  • E4.19: Let G be a group.
    (a) Show that if |G| ≤ 40, then G is solvable.
    (b) Show that if 41 ≤ |G| ≤ 59 or 61 ≤ |G| ≤ 89, then G is not simple.
    (c) What can you determine about the solvability or simplicity of G if |G| = 60?
    (d) Challenge problem: Show that if |G| = 90 then G is solvable.
    (Hint: You may use E4.16 in all parts of this problem.)
  • E4.20: Let G be the finite abelian group presented by G = ⟨ x,y | x¹² = 1, y¹⁸ = 1, xy = yx, x² = y³ ⟩. (That is, G is a quotient of the group H = ⟨ x | x¹² = 1 ⟩ × ⟨ y | y¹⁸ = 1 ⟩, over the subgroup N = ⟨ x²y^-3 ⟩.) Find the invariant factor decomposition of G.
  • E4.21: Prove that if G is an abelian group of order 2ⁿ with n ≥ 1, then the number of elements of G of order 2 is an odd number.
  • E4.22: An abelian group G, with group operation written as addition, is called divisible if for each g ∈ G and n ∈ N, there is a g' ∈ G such that g = ng'. (For example, the group Q of rational numbers is divisible.) Prove that if G is a nontrivial divisible abelian group, then G is not finitely generated.
  Due 11/17/17 for grading: E4.19(a), E4.20, E4.21

• Problem Set 11:
  • E5.1: Prove the following results from the TOC: (a) TOC Thm 5.37. (b) TOC Thm 5.50. (c) TOC Prop 5.58. (d) TOC Thm 5.64. (e) TOC Thm 5.67.
  • E5.2: An element r of a ring R is called nilpotent if r^m = 0 for some integer m ≥ 1.
    (a) Let X be a set and let R = Fun(X,R) be the ring of functions from X to R. Prove that R contains no nonzero nilpotent elements.
    (b) Prove that if R is a ring with 1 and b is a nilpotent element of R, then 1-b is a unit in R.
    (c) Let n be a natural number. Show that Z/nZ has no nonzero nilpotent elements if and only if n is squarefree. (An integer is squarefree if its prime factorization has no repeated primes.)
  • E5.3: The center of a ring R is the set {z ∈ R | zr = rz for all r ∈ R}.
    (a) Prove that the center of a ring with 1 is a subring that contains (the same) 1.
    (b) Determine the center of the real quaternion ring H.
  • E5.4: Let R be a ring with 1, and let p(x) = a₀ + a₁x + ... + a_nxⁿ be an element of R[x]. Prove that p(x) is a zero divisor in R[x] if and only if there is a nonzero b ∈ R with bp(x) = 0.
    (Hint: The text outlines a proof in problem 7.2.2 p. 238.)
  • E5.5: (a) Let R be the group ring ZS₃ and let a = e + (1 2), b = e - (1 2), c = 2e + 3(2 3) - (1 2), and d = 5(123). Compute the elements: (a-i) a³. (a-ii) ab. (a-iii) 3a - bcd.
    (b) Let R' be the group ring Z/3ZS₃. Viewing a,b,c,d from part (a) as elements of R', compute the three elements of R' listed in (a-i-iii).
    (c) Let G be a finite group and let R'' be the group ring ZG.
    (c-i) Prove that the element r = ∑_{g ∈ G} g is in the center of R''.
    (c-ii) Prove that if |G| is even, then R'' is not an integral domain.
  • E5.6: An element p of an integral domain R is called irreducible if p is not a unit and whenever p = xy for some x,y ∈ R, then either x or y is a unit. Two irreducible elements p and q are called associates if p = uq for some unit u of R. ("Associate" is an equivalence relation on the set of irreducible elements of R.)
    (a) Find the set of irreducible elements of Z, and determine which are associates of each other.
    (b) Let R be the integral domain Z[i] = {a + bi | a,b ∈ Z} (a subring of the complex numbers). Show that 13 = 13 + 0i is not an irreducible element of R. (Hint: 13 is the sum of two squares.)
  Due 12/1/17 for grading: E5.1(c), E5.2(b), E5.3(b), E5.5(c), E5.6

• Problem Set 12:
  • E5.7: Prove the following results from the TOC: (a) TOC Lemma 5.94. (b) TOC Thm 5.97. (c) TOC Thm 5.105 (LITR). (d) TOC Prop 5.114.
  • E5.8: Let I and J be ideals of a ring R with 1 ≠ 0.
    (a) Show that if R is commutative and I + J = R, then IJ = I ∩ J.
    (b) Show that if I = (a₁,...,a_m) and J = (b₁,...,b_n), then IJ = ({a_ib_j | 1 ≤ i ≤ m, 1 ≤ j ≤ n}).
  • E5.9: In each part, determine (and prove) whether or not the ring R is (ring) isomorphic to the ring S.
    (a) R = Z and S = Z[x].
    (b) R = Z[x]/((x²+1)Z[x]) and S = Z[i].
    (c) R = Z/3Z and S is the subring {t ∈ Z/6Z | 3t = 0} of Z/6Z. (The set S is actually an ideal of Z/6Z, called the annihilator of the element 3 of Z/6Z.)
    (d) R = Z/3Z × Z/3Z and S = (Z/3Z)[x]/I, where I = (x² + 1).
    (e) R = RG where G = Z/2Z × Z/2Z, and S = M₄(R).

• Exam 2 due Sunday December 10 6:00pm

  S. Hermiller.