Questions and Answers of Algebra Graduate Texts In Mathematics

The operation of free product is commutative and associative: for any groups A,B,C, A * B ≅ B * A and A * (B * C) ≅ (A * B) * C.
Find subgroups Hand K of D4* such that H ◁ Kand K ◁ D4*, but His not normal in D4 *.
Prove that an infinite group is cyclic if and only if it is isomorphic to each of its proper subgroups.
If α, β are cardinals, define αβ to be the cardinal number of the set of all functions B → A, where A,B are sets such that IAI = α, IBI = β(a) αβ is independent of the choice of A,B.(b)
The following conditions on a group Gare equivalent: (i) G is abelian; (ii) (ab)2 = a2b2 for all a,b ϵ G; (iii) (ab)-1 = a-1b-1 for all a, b ϵ G; (iv) (ab)n = anbn for all n c Z and all a,b ϵ G;
If I is an infinite set, and for each i ϵ / Ai is a finite set, then |U A₂| ≤ |I|. iel
Let G be a group of order 2n; then G contains an element of order 2. If n is odd and G abelian, there is only one element of order 2.
Proof that If G is a group, then C = {a ϵ G I ax = xa for all x ϵ G} is an abelian subgroup of G. C is called the center of G.
Let α be a fixed cardinal number and suppose that for every i ϵ I, Ai is a set with IAiI = α. Then IU A: ≤ 1a. iel
Find all normal subgroups of Dn. The value of n is infinity thats why we assume any amount.
Proof that If N is the normal subgroup of A * B generated by A, then (A * B)/ N ≅ B.
If His a cyclic subgroup of a group G and His normal in G, then every subgroup of His normal in G. [Compare Exercise 10.]Exercise 10Find subgroups Hand K of D4* such that H ◁ Kand K ◁
Let {Gi| i ϵ I} be a family of groups and J ⊂ I. The map α:given by {ai}|→ {bi}, where bi = ai for j ϵ J and bi = ei (identity of Gi·) for i ∉ J, is a monomorphism of groups and II G; →
If G is a group, a, b ϵ G and bab-1 = ar for some r ϵ N, then biab-i = aϒi for all j ϵ N.
A normal subgroup Hof a group G is said to be a direct factor (direct summand if G is additive abelian) if there exists a (normal) subgroup K of G such that G = H X K.(a) If H is a direct factor of K
The group D4* is not cyclic, but can be generated by two elements. The same is true of Sn (nontrivial). What is the minimal number of generators of the additive group Z⊕Z?
The center (Exercise 2.11) of the group Dn is (e) if n is odd and isomorphic toZ2 if n is even.Exercise 2.11Find all normal subgroups of Dn.
Proof that If G and H each have more than one element, then G *H is an infinite group with center (e).
Proof that If H is a normal subgroup of a group G such that Hand G/ H are finitely generated, then so is G.
For each n ≥ 3 let Pn be a regular polygon of n sides (for n = 3, Pn is an equilateral triangle; for n = 4, a square). A Symmetry of Pn is a bijection Pn -Pn that preserves distances and maps
Proof that If a2 = e for all elements a of a group G, then G is abelian.
If p > q are primes, a group of order pq has at most one subgroup of order p. Suppose H,K are distinct subgroups of order p. Show H ∩ K = (e).
If G = (a) is a cyclic group and His any group, then every homomorphism ∫: G → H is completely determined by the element ∫(a) ϵ H.
Prove that a free group is a free product of infinite cyclic groups.
(a) Let H ◁ G, K ◁ G. Show that H V K is normal in G.(b) Prove that the set of all normal subgroups of G forms a complete lattice under inclusion (Introduction, Exercise 7.2).Exercise
(a) The set of all subgroups of a group G, partially ordered by set theoretic inclusion, forms a complete lattice in which the g.1.b. of {Hi |i ϵ I} isHi and the l.u.b. is(b) Exhibit the lattice of
(a) Let G be a group and {Hi| i ϵ I} a family of subgroups. State and prove a condition that will imply thatHi is a subgroup, that is, that(b) Give an example of a group G and a family of subgroups
The following cyclic subgroups are all isomorphic: the multiplicative group (i) in C, the additive group Z4 and the subgroup 2 2 3 4 3 4 :)) of S4.
For i = 1, 2 let Hi ⊲ Gi and give examples to show that each of the following statements may be false:(a) G1 ≅ G2 and H1 ≅ H2 => G1/ H1 ≅ G2/ H1.(b) G1 ≅ G2 and G1/
If G is a finite group of even order, then G contains ~n element a ≠ e such that a2 = e.
If G is the group defined by generators a,b and relations a2 = e, b3 = e, then G ≅ Z2 * Z3.
Prove that if N1◁ G1, N2 ◁ G2 then (N1 X N2) ◁ (G1 X G2) and (G1 X G2)/(N1 X N2) ≅ (G1/ N2) X (G2/ N2).
Let G be a nonempty finite set with an associative binary operation such that for all a, b, c ϵ G ab= ac ⇒ b = c and ba = ca ⇒ b = c. Then G is a group. Show that this conclusion may be false if
Let a1, a, ... be a sequence of elements in a semigroup G. Then there exists a unique function ψ : N* → G such that ψ(1) = a1, ψ(2) = a1a2, ψ(3) = (a1a2)a3 and for n ≥ 1,ψ(n + 1) =
For each prime p the additive subgroup Z(p∞) Qf Q/Z (Exercise 1.10) is generated by the setData from in exercise 1.10 Let G be an abelian group and let H,K be subgroups of G. Show that the join HVK
Let G be a group and Aut G the set of all automorphisms of G.(a) Aut G is a group with composition of functions as binary operation.(b) Aut Z ≅ Z2 and A ut Z6 ≅ Z2; Aut Z8 ≅
If ∫: G1 → G2 and g: H1 → H2 are homomorphisms of groups, then there is a unique homomorphism h : G1 * H1 → G2 * H2 such that h|G1 = ∫ and h| H1 = g.
If ∫: G → H is a homomorphism, H is abelian and N is a subgroup of G containing Ker ∫, then N is normal in G.
Let G be an abelian group and let H,K be subgroups of G. Show that the join H V K is the set {ab| a ϵ H, b ϵ K}. Extend this result to any finite number of subgroups of G.
(a) Consider the subgroups (6) and (30) of Z and show that (6)/(30) ≅ Z5.(b) For any k, m > 0, (k)/(km) ≅ Zm; in particular, .Z/(m) = (1)/(m) ≅ Zm.
If ∫: G → H is a homomorphism with kernel N and K < G, then prove that ∫-1( ∫(K)) = KN. Hence ∫-1(∫(K)) = K if and only if N < K.
If N ◁ G, [G: N] finite, H < G, IHI finite, and [G: N] and IHI are relatively prime, then H < N.
If N◁ G, IN/ finite, H < G, [G: H] finite, and [G: H] and INI are relatively prime, then N < H.
If H is a subgroup of Z(p∞) and H ≠ Z(p∞), then Z(p∞)/ H ≅ Z(p∞).

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