1. I f A and B are normal subgroups of G such that G/A and G/B are abelian, prove that G/(A intersect B) is abelian

2. Let H and K be groups, let f be a homomorphism from K into Aut(H) and as usual identify H and K as subgroups of G= H x_f K( x_f denotes product of H and K under f).

Prove that C_K(H)= Ker(f)

ps. C_K(H) is centralizer

keywords: semi direct

2. Let H and K be groups, let f be a homomorphism from K into Aut(H) and as usual identify H and K as subgroups of G= H x_f K( x_f denotes product of H and K under f).

Prove that C_K(H)= Ker(f)

ps. C_K(H) is centralizer

keywords: semi direct

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