Let h :from G to G\' and g : from G to G\'\' be group homomorphisms, in which G,

ID: 2986238 • Letter: L

Question

Let h :from G to G' and g : from G to G'' be group homomorphisms, in which G, G'

and G'' are groups. Denote the identity elements of G, G' and G'' by e, e' and e'' respectively.

(1) True or false: (a) h(e) = e' ; (b) g(e') = e''.

(2) Prove that gh : from G to G'' is a group homomorphism.

(3) Prove that Ker(h) is included in Ker(gh).

(4) Prove that, for some a in G and n in Z, if a^n = e then (h(a))^n = e'.

Hint. For (2), let x; y in G and show gh(xy) = gh(x)gh(y).

For (3), how to show X Y in set theory? Here gh stands for the composite of the maps (so that gh(z) =?).

For (4), usethe assumption that h is a homomorphism while a^n = e (in particular, h(a^n) =?).

i hope it helps you

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