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

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) =?).

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Explanation / Answer

Identity
A homomophism maps the identity in G into the identity in G'.
Thus alpha(0) = 0', and beta(0) = 0', thus 0 is in H.
Inverse
A homomophism maps the inverse in G into the inverse in G'.
Thus for a in H, alpha(-a) +' alpha(a) = 0' and beta(-a) +' beta(a) = 0'. Since the inverse is unique in G', beta(-a) = alpha(-a) ==> -a is in H.
Closure
From the above 0' = alpha(-a) +' beta(a) for any a in H
For a,b in H, alpha(a+b) = alpha(a) +' alpha(b)
Using commutativity in G' freely
= (alpha(a) +' alpha(b)) +' 0'
= (alpha(a) +' alpha(b) +') +' (alpha(-b) +' beta(b))
= (alpha(a) +' alpha(b) +'alpha(-b)) +' beta(b))
= alpha(a) +' beta(b)
= 0' +' (alpha(a) +' beta(b))
= (beta(a) +' alpha(-a)) +' (alpha(a) +' beta(b))
= beta(a) +' beta(b)
Thus if a, b are in H, so is a+b.
Commutativity
For a, b, c in H they are also in G and commute in G and H.

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