Hi, I have a few questions on my abstract algebra homework that is bugging me, and wondered if anyone can send me some help!
1. Let G, G' and G'' be groups. let φ: G > G' and γ: G' > G' be homomorphisms. Prove γ(φ(x)) is a homomorphism from G to G''.
2. If G is an abelian group, prove that the mapping θ: G > G' defined by θ(x)=x^-1 is an automorphism.
3. Suppose φ:G > G' is an isomorphism from G to G'. Prove that if H is a subgroup of G, then φ(H) (the image set of H under the map) is a subgroup of G''.
any help would be appreciated, proofs always have me stumped!