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4 part problem regarding homomorphisms

AutGuy98

New member
Sep 11, 2019
20
Hey guys,

I have some more problems that I need help with figuring out what to do. The first one is divided into 4 mini-problems, sub-sections, whatever you would like to call them. It asks:

(a) Show that (Z/4Z,+) is not isomorphic to ((Z/2Z) x (Z/2Z),+). Find a homomorphism from (Z/4Z,+) to ((Z/2Z) x (Z/2Z),+).
(b) Let G and H be groups and let $\phi:G\to H$ be a homomorphism. Show that $ker(\phi)$ is a normal subgroup of G and that $\phi(G)$ is a subgroup of H.
(c) Find a pair of groups G,H and a homomorphism $\phi:G\to H$ between them such that $\phi(G)$ is not normal in H.
(d) Let G and G' (i.e. G prime) be finite groups whose orders have no common factor. Prove that the only homomorphism $\phi:G\implies G'$ is the trivial one (i.e. $\phi(x)=1$ for all x's in G).

I would greatly appreciate it if someone could please get back to me about these by tomorrow. But if more time is needed to work the problems out, I completely understand. Thank you in advance to whomever assists me with these. I am extremely grateful.
 
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