1.) show that if G, G', and G'' are groups and if Ø: G -> G' and ¥: G' -> G'' are homomorphisms, then the composite map ¥Ø:G-> G'' is a homomorphism.
2.) Let Ø:G-> H be a group homomorphism. show that Ø[G] is abelian if and only if for all x, y element of G, we have xyx^-1y^-1 element of Ker(Ø).
3.) Let G be any group and let a be any element of G.
Let Ø: Z-> G be defined by Ø(n)= a^n. show that Ø is a homomorphism. Describe the image and the possibilities for the kernel of Ø.
4.) let Ø: G-> G' be a homomorphism with kernel H and let a element of G. Prove the set equality {x є G l Ø(x) = Ø(a) } = Ha.
2007-09-20
19:46:47
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2 answers
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Anonymous