Selected Algebra Exercises

1 Groups

Exercise 1.1 Suppose G is a group such that x2 = 1 for all x ∈ G. Prove that G isabelian.

Exercise 1.2 Prove that if G is an infinite cyclic group with generator x then x andx-1 are the only generators of G.

Exercise 1.3 Let G be a group.

(a) Suppose x ∈ G has order k. Prove that for any m ∈ Z, xm = 1 if and only if k|m.

(b) Suppose G is abelian and finite of order n and x ∈ G. Prove that the order of xdivides n.

Exercise 1.4 Let H and K be subgroups of a group G, each of finite index. Provethat H ∩K has finite index.

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Exercise 1.5 Suppose G is a finitely generated abelian group, for which every gen-erator has finite order. Prove that G is finite.

Exercise 1.6 Suppose H ≤ G and the product of any two left cosets of H is a leftcoset of H. Prove that H is normal in G.

Exercise 1.7 Let G be a group. Prove that G is finite if and only if it has a finitenumber of subgroups.

Exercise 1.8 Let G be a group with H,K �G such that H ∩K = 1.

(a) Prove HK = KH.

(b) Suppose G = HK. Prove that G ∼= H ×K.

(c) Suppose G = HK and N � G such that N ∩ H = 1 = N ∩ K. Prove thatN ≤ Z(G).

(d) Suppose N1, N2, N3 are normal subgroups of G such that Ni∩Nj = 1 for all i 6= jand NiNj = G for all i 6= j. Prove that G is abelian and Ni

∼= Nj for all i, j.

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Exercise 1.9 Suppose G is a group and there are subgroups H ≤ G, N�G such thatG = NH and N ∩H = 1. Prove that there is a homomorphism ϕ : H −→ Aut(N)such that G ∼= N oϕ H.

Exercise 1.10 Let G be a finite group of order mn where m and n are relativelyprime. Let a, b ∈ G with order m and n, respectively. Prove that G is cylic if andonly if ab = ba.

Exercise 1.11 Let G and H be cyclic groups.

(a) Suppose G and H are finite. Prove that G×H is cyclic if and only if the ordersof G and H are relatively prime.

(b) Suppose G is infinite. Prove that G×H is cyclic if and only if H is trivial.

Exercise 1.12 Suppose G is a group and H is a normal subgroup of G such that

both H and G/H are cyclic. Prove that G is generated by two elements.

Exercise 1.13 Suppose G is a group and H � G such that G/H∼= Z. Prove that

there is a subgroup K ≤ G such that H ∩K = 1 and G = HK.

Exercise 1.14 Prove that it is impossible for a group to be the union of two propersubgroups and give an example of a group that is the union of three proper subgroups.

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Exercise 1.15 Let G be a group with exactly one nonidentity element g of order n.Prove that n = 2 and g ∈ Z(G).

Exercise 1.16 Let G be a finite group of even order.

(a) Prove that G has an odd number of elements of order two.

(b) Prove that G has an odd number of elements of even order.

Exercise 1.17 Let G be a group of order 2n where n is odd. Prove that G has anormal subgroup of order n.

Exercise 1.18 Suppose G is a group of order 2n containing exactly n elements oforder two. Let H ⊆ G be the set of n elements of G not of order two.

(a) Prove that n is odd and H is a normal subgroup of G.

(b) Suppse a, b ∈ G have order two. Prove that ab ∈ H, and if a 6= b then ab 6= ba.

(c) Prove that H is abelian.

(d) Prove if n > 1 then Z(G) = 1.

(e) Prove that G ∼= H o Z/2Z.

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Exercise 1.19 Prove that G = (Q,+)/(Z,+) is an infinite group, which has elements

of arbitrarily large finite order, but no element of infinite order. Give another exampleof a group with this property.

Exercise 1.20 Let G be a group and N �G such that N is cyclic. Prove that everysubgroup of N is normal in G.

Exercise 1.21

(a) Prove that the set of all inner automorphisms of a group G is a normal subgroupof Aut(G).

(b) Prove that the group of inner automorphisms of a group G is isomorphic toG/Z(G).

(c) Let G be a group and H a subgroup. Prove that Aut(H) has a subgroup isomor-

phic to NG(H)/CG(H).

Note Part (b) implies part (a) with H = G.

Exercise 1.22 For σ ∈ Sn, the cycle type of σ is a tuple (i1, . . . , in) ∈ Nn suchthat the disjoint cycle decomposition of σ contains exactly ij cycles of lenth j, for1 ≤ j ≤ n. Prove that σ, τ ∈ Sn are conjugate if and only if they have the same cycletype.

Exercise 1.23

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(a) Suppose G and H are groups such that ϕ : G −→ H is an isomorphism. LetK �H. Prove that ϕ-1(K) �G and

G

ϕ-1(K)∼=H

K

(b) Suppose J �G and K �H. Prove that

G×HJ ×K

∼=G

J× H

K

Exercise 1.24 Prove that there is no homomorphism from S5 onto a group of order24.

Exercise 1.25 LetG be a group and f : G −→ G a function such that f(a)f(b)f(c) =f(x)f(y)f(z) for any abc = xyz = 1. Prove that there is some g ∈ G so thath : G −→ G with h(x) = gf(x) is a homomorphism.

Exercise 1.26

(a) Suppose G is an abelian group such that every nontrivial element has order p forsome fixed prime p. Prove that

G ∼=⊕κ

Z/pZ

for some cardinal κ.

(b) Suppose G and H are abelian groups such that every nontrivial element has or-der p for some fixed prime p. Prove that |G| = |H| implies G ∼= H.

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Exercise 1.27 Let G be an abelian group and suppose x, y ∈ G such that |x| = mand |y| = n. Prove that there exists z ∈ G such that |z| = lcm(m,n).

Exercise 1.28 Let G be a finite abelian group.

(a) Suppose |G| = mn where m and n are relatively prime. Prove that there aresubgroups H,K ≤ G of orders m and n respectively such that G = HK.

(b) Suppose H,K ≤ G with |H| = m and |K| = n. Prove that G has a subgroup oforder lcm(m,n).

Exercise 1.29 Let G be a finite abelian group. Prove that G is cylic if and only iffor all n > 0 there are at most n elements g ∈ G such that gn = 1.

Exercise 1.30 LetG be a group and suppose there is g ∈ G such that CG(g) = Z(G).Prove that G is abelian.

Exercise 1.31 Let G be a group and suppose G/Z(G) is cyclic. Prove G is abelian.

Exercise 1.32 Let G be a group. Let G′ be the commutator subgroup.

(a) Prove that if H ≤ G such that G′ ≤ H, then H�G. Thus, in particular, G′�G.

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(b) Prove that G/G′ is abelian.

(c) Prove that if H �G such that G/H is abelian, then G′ ≤ H.

Exercise 1.33

(a) Let G be an abelian group with odd order. Prove that the product of all theelements in G is 1.

(b) Let G be a group with odd order. Prove that the product of all elements (in anyorder) is an element of G′.

Exercise 1.34 Let G be a cyclic group.

(a) Prove that if the order of G is infinite then Aut(G) ∼= Z2.

(b) Prove that if the order of G is prime, then Aut(G) = Zp−1.

Exercise 1.35 Let n be a positive integer.

(a) Prove that Aut(S2) is trivial.

(b) Suppose n > 2. Prove that Z(Sn) is trivial.

(c) Suppose ϕ ∈ Aut(Sn). Prove that ϕ ∈ Inn(Sn) if and only if ϕ sends transposi-tions to transpositions.

(d) Suppose n 6= 6. Prove that Aut(Sn) = Inn(Sn).

(e) Suppose n > 2 and n 6= 6. Prove that Aut(Sn) ∼= Sn.

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(f) Prove that [Aut(S6) : Inn(S6)] ≤ 2.Remark: In fact, outer automorphisms of S6 do exist. Part (a) and Ex-ercise 1.21 imply that Inn(S6) = S6 and it can be shown that Aut(S6) ∼=Inn(S6) o Z/2Z, so

Aut(S6) ∼= S6 o Z/2Z.

Exercise 1.36 Let G be a group with |G| > 2. Prove that Aut(G) is nontrivial.

Exercise 1.37

(a) Let G be a group, H a proper subgroup of finite index n. Prove that G has aproper normal subgroup of finite index dividing n!, which is the largest normalsubgroup of G contained in H.

(b) Prove that if G is a simple group with a proper subgroup of index n, then |G|divides n!. Therefore an infinite simple group cannot have a proper subgroupof finite index.

(c) Suppose G is a finite group and p is the smallest prime dividing |G|. Let H ≤ Gsuch that [G : H] = p. Prove that H �G.

Exercise 1.38 Suppose G is a finite group and H is a normal subgroup of G of orderp, for some prime p.

(a) Prove that if gcd(|G|, p− 1) = 1, then H ≤ Z(G).

(b) Prove that if p is the least prime dividing |G|, then H ≤ Z(G).

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Exercise 1.39 Let G be a finitely generated group with a unique maximal subgroupH. Prove that G is cyclic of order pn for some prime p.

Exercise 1.40

(a) Let G be a finite abelian group and suppose g ∈ G has maximal order. Provethat there is a subgroup K ≤ G such that 〈g〉 ∩K = 0 and G ∼= 〈g〉+K.

(b) Given an example where part (a) is false when G is not assumed to be abelian.

Exercise 1.41 Suppose H is a normal subgroup of G of prime index p. Prove thatfor all K ≤ G, either K ≤ H or G = HK and [K : K ∩H] = p.

Exercise 1.42 Suppose G is a finite group, p is a prime, and H�G such that [G : H]is relatively prime p. Prove that H contains every element of G with p-power order.

Exercise 1.43

(a) Suppose G is a group, K is a normal subgroup of G, and H is a subgroup of Gcontaining K. If N = NG(H), prove that

NG/K

(H/K

)= N/K.

(b) Suppose G is a finite p-group, and H is a proper subgroup. Prove that H is aproper subgroup of N = NG(H).

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Exercise 1.44 Let G be a group acting on a set S and suppose N is a normalsubgroup of G. Let SN = {x ∈ S : ax = x ∀ a ∈ N}. Prove that g(SN) = SN for allg ∈ G.

Exercise 1.45 Let · be an action of a group G on a set A. For a ∈ A, let Oa ={g · a : g ∈ G} be the orbit of a, and let Sa = {g ∈ G : g · a = a} be the stabilizer ofa in G.

(a) Prove that Sa is a subgroup of G.

(b) Prove that |Oa| = [G : Sa] and if A′ is a set of representatives for {Oa : a ∈ A}then

|A| =∑a∈A′

|Oa|.

(c) Suppose G is a finite p-group acting on a set A. Let A0 = {a ∈ A : g ·a = a ∀ g ∈G}. Prove that

|A| ≡ |A0| (mod p).

Exercise 1.46 Let G be a group with Z(G) the center of G. Suppose g1, . . . , gk arerepresentatives for the distinct conjugacy classes of elements not in Z(G) then

|G| = |Z(G)|+k∑i=1

[G : CG(gi)],

where CG(gi) = {x ∈ G : xgi = gix} is the centralizer of gi in G.

Exercise 1.47

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(a) If G is a group and |G| = pn for some prime p and n > 0, then Z(G) is nontrivial.

(b) Suppose G is as in part (a). If {e} 6= N �G, then N ∩ Z(G) is nontrivial.Note: This generalizes part (a).

(c) Given a prime p, let G be the set of infinite upper triangular matrices with 1

along the diagonal and arbitrary elements of Z/pZ above the diagonal with only

finiitely many entries nonzero. Prove that under matrix multiplication modulop, G is an infinite p group with trivial center.

Exercise 1.48 Let H be a subgroup of a group G. Prove that [G : NG(H)] is equalto the number of conjugates of H in G.

Exercise 1.49 Let G be a finite group.

(a) If G is cyclic and d divides |G| then there is x ∈ G such that |x| = d.

(b) If G is abelian and p divides |G| then prove there is x ∈ G such that |x| = p.

Exercise 1.50 Let G be a finite group.

(a) If |G| = pn then prove G has a subgroup of order pk for 1 ≤ k ≤ n.

(b) If |G| = pnm where p and m are relatively prime then prove G has a subgroupof order pn.

(c) If pn divides |G| for some n then prove G has a subgroup of order pn.

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(d) If p divides |G| then prove G has an element of order p. (Compare to 1.49b.)

(e) Prove that every finite p-group is solvable.

Exercise 1.51 Suppose G is a finite group, P is a Sylow p-subgroup of G, and H isa p-subgroup of G.

(a) Prove that there is some g ∈ G such that H ≤ gPg-1. In particular, any twoSylow p-subgroups of G are conjugate.

(b) Suppose H is normal in G. Prove that H ≤ P .

(c) Let np,H be the number of Sylow p-subgroups of G containing H, and let np =np,{1} be the number of Sylow p-subgroups of G. Prove that np,H ≡ np (mod p).

(d) With np as in part (c), prove that np ≡ 1 (mod p) and if |G| = pnm where p6 |m,then np|m.

Exercise 1.52 Let G be a finite group and P a Sylow p-subgroup of G.

(a) Prove that P is the only Sylow p-subgroup of G contained in NG(P ).

(b) Let H = NG(P ). Prove that NG(H) = H.

(c) Suppose that K �H �G and K is a Sylow p-subgroup of H. Prove that K �G.(Compare to Exercise 1.68e)

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Exercise 1.53 Suppose G is a finite group, H �G, and K is a Sylow p-subgroup ofH. Prove that G = NG(K)H.

Exercise 1.54 Let H be a Sylow p-subgroup of G and let K�G. Prove that H ∩Kis a Sylow p-subgroup of K.

Exercise 1.55 Prove that a Sylow p-subgroup of D2n is cyclic and normal for everyodd prime p.

Exercise 1.56 Let G be a finite group let H be the subgroup of G generated by all

elements of odd order. Prove that H is normal in G and∣∣∣G/H ∣∣∣ is a power of 2.

Exercise 1.57 Let G a finite group and suppose p is a prime dividing |G|.

(a) Let P be a Sylow p-subgroup of G. Prove that if x ∈ NG(P ) and xp = 1 thenx ∈ P .

(b) Let X = {x ∈ G : xp = 1}. Prove that p divides |X|.

Exercise 1.58 Suppose G is a finite group and H a proper subgroup of G. Provethat G is not the union of the conjugates of H.

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Exercise 1.59 Let G be a group such that for any two nontrivial elements a, b ∈ G,there is an automorphism of G sending a to b.

(a) Prove that all nontrivial elements of G have the same order. Furthermore, ifsome nontrivial element of G has finite order, then there is a prime p such thatall nontrivial elements of G have order p.

(b) Prove that if G is finite then G is abelian.

Exercise 1.60 Let G be a finite group with H ≤ G and N �G.

(a) Suppose (|H|, [G : N ]) = 1. Prove that H ≤ N .

(b) Suppose (|N |, [G : H]) = 1. Prove that N ≤ H.

Exercise 1.61 Let G be a group.Definition If s is an ordered set and {Hi : i ∈ s} is a collection of normal subgroupsof G, we define

⊙i∈s

Hi =

{∏i∈s

xi : xi ∈ Hi, xi = 1 for all but finitely many i

},

where multiplication is done consistent with the ordering of s.

(a) If s is an ordered set and {Hi : i ∈ s} is a collection of normal subgroups of Gthen prove that

⊙i∈sHi is a subgroup of G.

(b) Suppose κ is an ordinal and {Hi : i ∈ κ} is a collection of normal torsionsubgroups of G such that for any distinct i, j we have gcd(|x|, |y|) = 1 for allx ∈ Hi and y ∈ Hj. Prove that for any i ∈ κ and any finite s ⊆ k with i 6∈ s,we have

Hi ∩⊙j∈s

Hj = 1.

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(c) Suppose κ is an ordinal and {Hi : i ∈ κ} is a collection of normal subgroups ofG such that G =

⊙i∈κHi and for all i ∈ κ and finite s ⊆ κ with i 6∈ s we have

Hi ∩⊙

j∈sHj = 1. Prove that G ∼=⊕

i∈κHi.

(d) Give an example of a group G with H,K subgroups of G such that H ∩K = 1and G = HK but G 6∼= H ×K.

Exercise 1.62 Let G be a torsion group such that each Sylow p-subgroup of G isnormal in G.

(a) Prove that G is isomorphic to the direct sum of its Sylow subgroups.

(b) Prove that if H is a Sylow subgroup of G then Z(H) = Z(G) ∩H.

Exercise 1.63 If G is group of order p2 for some prime p then G is abelian.

Exercise 1.64 Let G be non-abelian group of order p3. Prove that Z(G) = G′.

Exercise 1.65 Let G be a finite group of order pq, where p, q are prime, p < q, andp does not divide q − 1. Prove that G is cyclic.

Exercise 1.66 Let p, q be primes. Let G be a group of order p2q

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(a) Prove that G is not simple.

(b) Suppose q < p and q does not divide p2 − 1. Prove that G is abelian.

Exercise 1.67 Let G be a finite group with exactly two nontrivial proper subgroups.Prove that one of the following must hold.

(i) G is cyclic of order pq for distinct primes p, q

(ii) G is cyclic of order p3 for some prime p.

(iii) G is isomorphic to Z/pZ ×Z/pZ for some prime p.

Exercise 1.68 A subgroup H of a group G is called characteristic (written H�G)if ϕ(H) ⊆ H for every ϕ ∈ Aut(G).

(a) Prove that if H �G then ϕ(H) = H for all ϕ ∈ Aut(G).

(b) Give an example where H ≤ G and ϕ ∈ Aut(G), but ϕ(H) ( H.

(c) Prove that every characteristic subgroup is normal.

(d) Prove that the converse of (c) is false.

(e) Give an example where K �H �G but K 6� G. Compare to Exercise 1.52c.

(f) Suppose K �H �G. Prove that K �G.

(g) Prove that the center of a group is a characteristic subgoup.

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(h) Prove that the commutator subgroup of a group is a characteristic subgroup.

Exercise 1.69

(a) Let G be a finite simple group whose order is divisible by at least two primes.Let p be a prime divisor of |G| and let np be the number of Sylow p-subgroupsof G. Prove that |G| divides np!.

(b) Prove that no group of order 1,000,000 is simple.

Exercise 1.70 Let G and H be groups and suppose ϕ : G −→ H and ψ : H −→ Gare group homomorphisms such that ψ ◦ ϕ(g) = g for all g ∈ G.

(a) Prove that ϕ is injective and ψ is surjective.

(b) Prove that H = ϕ(G) kerψ.

(c) Suppose ϕ(G) is normal in H. Prove that H ∼= ϕ(G)× kerψ.

Exercise 1.71 Suppose

1 −−−→ Nι−−−→ G

ϕ−−−→ K −−−→ 1

is an exact sequence of groups. Suppose also that there is a group homomorphismτ : G −→ N such that τ ◦ ι = idN . Prove that G ∼= N ×K.

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Exercise 1.72 Let G be a group and fix a set of generators X. Let H be a subgroupof G and fix a set T of right coset representatives for H in G, with 1 ∈ T . For g ∈ G,let λ(g) ∈ T such that Hg = Hλ(g).

(a) Given a, b ∈ G, define (a, b) = λ(a)bλ(ab)-1. Prove that H is generated by ele-ments of the form (t, x), for t ∈ T and x ∈ X ∪X -1.

(b) Suppose G is finitely generated and H has finite index in G. Prove that H isfinitely generated.

Exercise 1.73 Let G be a group and H a subgroup of finite index n. Let {g1, . . . , gn}be a set of right coset representatives for H in G. If g ∈ G, then for 1 ≤ i ≤ n thereis some hi ∈ H and 1 ≤ ji ≤ n such that gig = higji . Note that the choice of hi andji is unique. We define the map

τH : G −→ H/H ′ such that τH(g) = h1 . . . hnH′.

Prove that τH does not depend on the choice of coset representatives, and that τH isa homomorphism (called the transfer homomorphism of H).

Exercise 1.74 Let G be a group and suppose Z(G) has finite index in G.

(a) Prove that G′ is finite.

(b) Suppose G is torsion free. Prove that G is abelian.

Exercise 1.75 If p is an odd prime and G ≤ Sp such that 2p divides |G| then G = Sp.

Exercise 1.76 Let G be a finite group and suppose k divides |G|. Consider thefollowing two statements.

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1. G has a subgroup of order k.

2. G has a subgroup of index k.

Prove that if G is abelian then both statements are true. Provide counterexamplesshowing that neither statement holds in general.

Exercise 1.77

(a) Give an example of two nonisomorphic finite groups of the same order each ofwhich contains the same number of elements of any given order.

(b) Suppose G and H are nonisomorphic finite groups of the same order containingthe same number of elements of any given order. Prove that S3×G and S3×Hare nonisomorphic groups containing the same number of elements of any givenorder.

Exercise 1.78 Given an abelian group G and n > 0, let Gn = {x ∈ G : nx = 0}.

(a) Suppose G and H are finite abelian p-groups such that |Gpn| = |Hpn| for alln ∈ N. Prove that G ∼= H.

(b) Let G and H be finite abelian p-groups. Then |Gpn| = |Hpn| for all n ∈ N ifand only if for all n the number of elements of G of order pn is the same as thenumber of elements of H of order pn.

(c) Let G and H be finite abelian groups such that for all m ∈ Z+ the number ofelements of G of order m is equal to the number of elements of H of order m.Then G ∼= H.

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Exercise 1.79

(a) Suppose G and H are finite abelian p-groups such that |G| = |H|, exp(G) =exp(H), and |{g ∈ G : pg = 0}| = |{h ∈ H : ph = 0}|. Provide a counterexam-ple to the claim that G ∼= H.

(b) Let p be prime. Find, with proof, the largest integer n such that the followingstatement is true.

If G and H are finite abelian p-groups such that |G| = |H| < pn, exp(G) =exp(H), and |{g ∈ G : pg = 0}| = |{h ∈ H : ph = 0}|, then G ∼= H.

Exercise 1.80 Let F be the set of multiplicative functions (i.e., f : Z+ −→ Csuch that f(1) = 1 and f(mn) = f(m)f(n) for all m,n relatively prime). Givenmultiplicative functions f, g, define the Dirchlet convolution, f ∗ g, of f and g by

f ∗ g(n) =∑d|n

f(d)g(nd)

Let ε be the multiplicative function such that ε(n) = 0 for all n > 1.

(a) Prove that F is an abelian group under ∗ with identity element ε.

(b) Prove that F is torsion free.

(c) For any prime p, let Fp = {f ∈ F : f(n) = 0 ∀ n > 0 st p6 |n}. Prove that Fp isa subgroup of F .

Exercise 1.81

Let C∗ denote the multiplicative group of nonzero complex numbers.

(a) Let n ∈ Z+. Describe all of the subgroups of C∗ of order n.

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(b) Suppose G is a finite group and ϕ : G −→ C∗ is a nontrivial homomorphism.Prove that ∑

x∈G

ϕ(x) = 0.

Exercise 1.82

(a) Prove that (Q,+) cannot be the direct sum of two proper subgroups.

(b) Prove that any finitely generated subgroup of (Q,+) is cyclic.

(c) Prove that (Q,+) is not finitely generated.

(d) Prove that (Q,+) is not a free abelian group.

Exercise 1.83 Prove that (Q+, ·) is a free abelian group and give the structure of(Q∗, ·).

Exercise 1.84 Consider the additive group Q/Z.

(a) Prove that the Sylow p-subgroup of Q/Z isZ[1

p]/Z.

(b) Prove that

Q/Z ∼=⊕p prime

Z[1p]/Z.

Exercise 1.85 Suppose G is an abelian group and H ≤ G is divisible.

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(a) Prove that there is a subgroup K ≤ G such that H ∩K = {0} and G = H +K.

(b) Prove that G ∼= G/H ×H.

Exercise 1.86 Let G be a divisible abelian group and let Tor(G) ⊆ G be the set

of elements of finite order. Prove that Tor(G) is a subgroup of G and G/Tor(G) is

torsion free.

Exercise 1.87 Let G be a torsion free divisible abelian group. Prove that G is avector space over Q and therefore there is some cardinal κ such that

G ∼=⊕κ

Q.

Exercise 1.88 Let G be an abelian group and suppose x1, x2, x2, . . . are nonzeroelements of G such that px1 = 0, px2 = x1, px3 = x2, . . .. Let H = 〈x1, x2, x3, . . .〉.

Prove that H ∼= Z[1p]/Z.

Exercise 1.89 Let p be a fixed prime.

(a) Prove that following groups are isomorphic.

(i) The group, under multiplication, of (pn)th roots of unity for all n ∈ N; i.e.{e

2πmipn : m,n ∈ Z+

}.

(ii) The Sylow p-subgroup of Q/Z; i.e.Z[1

p]/Z.

(iii) 〈x1, x2, x3, . . . : px1 = 0, px2 = x1, px3 = x2, . . .〉

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(iv) The direct limit of Z/pZp2→ Z/p2Z

p3→ Z/p3Zp4→ . . ., where pk denotes

multiplication by p modulo pk.

This group is called the Prufer p-group and denoted by Z(p∞). It is clearlyan infinite torsion group.

(b) Prove that for all n ∈ Z+, Z(p∞) has a unique cyclic subgroup of order pn. More-over, prove these are all of the proper, nontrivial subgroups of Z(p∞).

(c) Prove that Z(p∞) is divisible.

Exercise 1.90 The structure theorem for divisible abelian groups states that if G isa divisible abelian group then there are cardinals κ0 and κp for p prime such that

G ∼=⊕κ0

Q×⊕p prime

⊕κp

Z(p∞).

Let (κp)p=0, p prime be the signature of G. Prove that two divisible abelian groups Gand H are isomorphic if and only if they have the same signature.

Exercise 1.91 Suppose G, H, and K are either all finitely generated abelian groupsor all divisible abelian groups. Prove the following:

(a) G×G ∼= H ×H implies G ∼= H.

(b) G×K ∼= H ×K implies G ∼= H.

Exercise 1.92 Let C∗ be the multiplicative group of nonzero complex numbers.

(a) Give the divisible abelian group structure of C∗.

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(b) Let U ≤ C∗ be the subgroup of complex numbers of modulus 1. Prove that

U ∼=(R/Z,+

), and give the divisible abelian group structure of U .

(c) Conclude that C∗ ∼= U .

2 Rings

Exercise 2.1 Let R be a commutative ring with unity. Let I, J be ideals in R. Let(I : J) = {r ∈ R : rJ ⊆ I}. Prove that if I is primary and if J is not a subset of Ithen (I : J) is primary and

√(I : J) =

√I.

Exercise 2.2 Let R be a commutative ring with unity.

(a) Prove that maximal ideals are prime.

(b) Prove that if I is an ideal of R then R/I is an integral domain iff I is prime.

(c) Prove that if I is an ideal of R then R/I is a field iff I is maximal.

(d) Prove that the union of all maximal ideals of R is the set of non-units of R.

(e) Prove that R has a unique maximal ideal iff the set of non-units is an ideal.

(f) Prove that the union of all prime ideals of R is the set of non-units of R.

(g) Prove that the intersection of all prime ideals is the set of nilpotent elements of R.

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Exercise 2.3 Let R be a commutative ring such that R has a field F as a subring.Furthermore assume that R is a 2-dimensional F -vector space.

(a) Prove there is a ∈ R such that R = F [a].

(b) Prove that R ∼= F [x]/(p) where p(x) ∈ F [x] is quadratic.

(c) Prove that R is either a field or isomorphic to F ×F or isomorphic to F [x]/(x2).

(d) Prove that F × F and F [x]/(x2) are not isomorphic and neither is a field.

Exercise 2.4 Prove that for p prime, the cyclotomic polynomial Φp(x) =xp − 1

x− 1is

irreducible over Z.

Exercise 2.5 Prove that a finite integral domain D is a field.

Exercise 2.6 Let I be a nonzero ideal of Z[i]. Show that Z[i]/I is finite.

Exercise 2.7 Let R be a ring such that x4 = x for all x ∈ R. This exercise outlinesa proof that R is commutative.

(a) Prove that 2x = 0 for all x ∈ R.

(b) Prove that if a ∈ R such that a2 = a then (ax− axa)2 = 0 = (xa− axa)2.

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(c) Prove that if b ∈ R such that b2 = 0 then b = 0.

(d) Prove that xw3 = w3x for all x,w ∈ R.

(e) Prove that x(u+ u2) = (u+ u2)x for all x, u ∈ R.

(f) Prove that R is commutative. Hint : Apply part (e) with u = x+ y.

Exercise 2.8

(a) Suppose R is a ring with unity and x ∈ R is nilpotent. Prove that 1+x is a unit.

(b) Suppose x, y ∈ R are nilpotent. Prove that x+ y is nilpotent.

(c) Suppose R is a commutative ring. Prove that f(x) = anxn+. . .+a1x+a0x ∈ R[x]

is a unit if and only if a0 is a unit in R and ai is nilpotent for all 0 < i ≤ n.

3 Fields

Exercise 3.1 Let F be a field.

(a) Prove the division algorithm for F [x], i.e. if f(x), g(x) ∈ F [x] with g(x) nonzerothen there are p(x), r(x) ∈ F [x] such that f(x) = p(x)g(x) + r(x) and r(x) ≡ 0or deg(r) < deg(g).

(b) Prove that F [x] is a principal ideal domain.

(c) Suppose f(x) ∈ F [x] and c ∈ F . Prove that there is g(x) ∈ F [x] such thatf(x) = (x− c)g(x) + f(c).

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(d) Prove that if f(x) ∈ F [x] is nonzero of degree n then f(x) has at most n rootsin F .

Exercise 3.2 Suppose F is a finite field. Prove that F is not algebraically closed.

Exercise 3.3 Let F be a field, f(x) ∈ F [x] a quartic with four distinct rootsx1, x2, x3, x4 in an algebraic closure of F . Let K = F (x1, x2, x3, x4). Let

a = x1x2 + x3x4 b = x1x3 + x2x4 c = x1x4 + x2x3

Let E = F (a, b, c). Show that Γ(E/F ) ∼= G/G ∩H where G = Γ(K/F ) and

H = {1, (1 2)(3 4), (1 3)(2 4), (1 4)(2 3)} ⊆ S4

Exercise 3.4 Prove that any finite subgroup of the multiplicative group F ∗ of a fieldF is cyclic.

Note: This says that if F is a finite field then F ∗ is cyclic and so there is only onefinite field of a given order up to isomorphism.

Exercise 3.5 Let p be a prime and n ∈ Z+ such that there is a prime q such thatq|n and q26 |n. Prove that the Galois group of xp − n over Q is a semidirect productof cyclic subgroups.

Exercise 3.6 Let F be a field of characteristic p.

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(a) Prove that the map ϕ : F −→ F such that ϕ(a) = ap is an injective endomor-phism of K.

(b) Let f(x) = xp − x − a ∈ F [x]. Let α be a root of f(x) in a splitting field K off(x). Prove that K = F (α).

(c) If f(x) is irreducible over F , prove that AutF (K) ∼= Zp.

Exercise 3.7 Let K/F be a finite separable field extension. Prove that there areonly finitely many fields E with F ⊆ E ⊆ K.

Exercise 3.8 Let f(x) be an irreducible polynomial in Q[x] of prime degree p. Sup-pose f(x) has exactly p − 2 real roots. Let K be the splitting field of f(x) over Q.Show that Γ(K/Q) ∼= Sp.

Exercise 3.9 Let c ∈ Q and assume a3 6= c for all a ∈ Q. Prove that the Galoisgroup of x3 − c over Q is isomorphic to S3.

Exercise 3.10 Let c ∈ Q and assume a = 1 + c2 is not a square in Q. Prove thatQ(√a+√a/Q) is a Galois extension with cyclic group of order 4.

Exercise 3.11 Let f(x) = x8 +x5 +x4 +x3 +1, g(x) = x3 +1 ∈ F [x] where F = F2.

Determine a representative of the multiplicative inverse of g (modf) in F [x]/(f).

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Exercise 3.12 Suppose that F is a subfield of the field K.

(a) Suppose that [K : F ] = 2. Prove that K is a normal extension of F .

(b) Give an example where [K : F ] > 2 and K is not normal over F .

(c) Give an example where [K : F ] = 2 and K is not Galois over F .

Exercise 3.13 Let s1, s2, . . . , sn we the elementary symmetric functions in the inde-terminates x1, x2, . . . , xn. Show that Q(x1, . . . , xn) is a Galois extension of Q(s1, . . . , sn)of degree n! and that the Galois group is the symmetric group Sn.

Exercise 3.14 Let F be a finite extension of Q.

(a) If α ∈ F prove that there is an integer k ∈ Z+ such that kα is integral over Q.

(b) Given the Primitive Element Theorem, prove that F has an integral primitiveelement over Q.

Exercise 3.15 A field F is formally real if -1 cannot be written as a sum of squares.F is real closed if F is formally real with no proper formally real algebraic extensions.Prove that if F is a formally real field then F is contained in some real closed field.

Exercise 3.16 Suppose F is a formally field and let A be the set of elements of Fthan can be written as a sum of squares.

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(a) Prove that no sum of nonzero squares in F is zero, and so for all a ∈ F\{0}, atmost one of a or -a is in A.

(b) Suppose F is real closed. Prove that A is the set of squares in F , and for alla ∈ F either a or -a is in A.

Exercise 3.17 Let F be a formally real field. Prove that the following are equivalent.

(i) F is real closed.

(ii) For any a ∈ F , either a or -a is a square and every polynomial of odd degree hasa root.

(iii) F (i) is algebraically closed, where i2 = -1.

Exercise 3.18 A field F is orderable if there is a linear ordering < which respectsmultiplication and addition. Prove that F is orderable if and only if F is formallyreal.

Exercise 3.19 Let F be a real closed field with ordering <. Prove that for alla, b ∈ F , a < b if and only if b − a is a nonzero square and so, in particular, theordering of a real closed field is unique.

Exercise 3.20 Let F be a formally real field and set A to be the elements of F thatcan be written as a sum of squares.

(a) Let a ∈ F . Prove that F (√a) is formally real if and only if -a ∈ F\A. In

particular, exactly one of F (√a) and F (

√-a) is formally real.

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(b) Suppose -a ∈ F\A. Prove that there is an ordering of F such that a > 0.

Exercise 3.21 Let F be an ordered field. Prove that F is real closed if and only ifF satisfies the intermediate value property, i.e., for all f(x) ∈ F [x] and a, b ∈ F suchthat f(a) < 0 < f(b) there is some c between a and b such that f(c) = 0.

Exercise 3.22 Let F be a real closed field. Given f(x) = anxn+. . .+a1x+a0 ∈ F [x],

define the formal derivative f ′(x) = nanxn−1 + . . .+ 2a2x+ a1 ∈ F [x].

(a) (Rolle’s Theorem) Let f(x) ∈ F [x] and a, b ∈ F with a < b and f(a) = 0 = f(b).Prove that f ′(x) has a root in (a, b).

(b) (Mean Value Theorem) Let f(x) ∈ F [x] and a, b ∈ F with a < b. Prove thatthere is some c ∈ (a, b) such that

f ′(c) =f(b)− f(a)

b− a.

Exercise 3.23 Let x and y be algebraically independent over R. Show that R(x, y)is formally real and that we can find orders <1 and <2 of R(x, y) such that x <1 yand y <2 x.

Exercise 3.24

(a) Show that for q ∈ Q we can order Q(t) such that t− q is a positive infinitesimal,with respect to the ordering on Q.

(b) Show that we can order Q(t) such that t is infinite.

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4 Modules

Exercise 4.1 Let R be a ring, M an R-module and N a submodule. Prove that M

is Noetherian iff N and M/N are Noetherian.

Exercise 4.2 Let R be a commutative ring with unity and let G be the group ofR-module automorphisms of R. Show that G is isomorphic to the group R∗ of unitsof the ring R.

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