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Ten problems related to group theory from math 511a homework. The problems cover topics such as subgroups, homomorphisms, conjugacy, center of a group, automorphisms, and normal subgroups.
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If H and K are finite subgroups of a group G and if #H, #K are relatively prime, prove that H ∩ K = 1. (Here 1 denotes the trivial subgroup of G.)
Prove that A 4 and D 12 are non-isomorphic groups of order 12.
(i) If f : G → H is a homomorphism and x ∈ G has order k, prove that f (x) ∈ H has order dividing k. (ii) If #G, #H are relatively prime and f : G → H is a homomorphism, prove that f (x) = 1 for all x ∈ G.
(i) Prove that Z(GL 2 (F)) is the set of scalar matrices. (ii) If G is a group and G/Z(G) is cyclic, prove that G is abelian.
(i) If g ∈ G, prove that the map γg : G → G defined by γg (x) = gxg−^1 is an automorphism of G. (ii) Prove that the function Γ : G → Aut(G) with Γ(g) = γg is a homomor- phism. (iii) Prove that ker(Γ) = Z(G). (iv) Define Inn(G) = im(Γ), the group of inner automorphisms of G. Prove that Inn(G) C Aut(G).
2 MATH 511A, HOMEWORK 7
Gd = {g ∈ G : ord(g) | d}.
(i) Prove that Gd is a subgroup of G, and that Gm ∩ Gn = { 0 }. (ii) Prove that G = Gm + Gn = {g + h : g ∈ Gm and h ∈ Gn}. (iii) Prove that G ∼= Gm × Gn.