2. Let G and H be groups, and let a: G → H be a homomorphism. Prove the followingstatements.(a) a(G) is a subgroup of H.(b) If G is abelian, then a(G) is abelian.(c) If G is cyclic, then a(G) is cyclic.(d) If a is injective, then g and a(g) have the same order for any g = G.

Answered: 2. Let G and H be groups, and let a: G…

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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2. Let G and H be groups, and let a: G H be a homomorphism. Prove the following statements. (a) a(G) is a subgroup of H. (b) If G is abelian, then a(G) is abelian. (c) If G is cyclic, then a(G) is cyclic. (d) If a is injective, then g and a(g) have the same order for any g € G.