(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

Let A, B be groups and theta: A --> Aut(B) a homomorphism. For a in A denote theta(a)= theta_a in Aut(B). Equip the product set B x A={(b,a): a in A, b in B} with the binary operation (b,a)(b',a')= (b'',a'') where a''=aa' and b''=b(theta_a{b')).

Show that this binary operation induces a group structure on the set B x A (ie it satisfies the group axioms).

3. The attempt at a solution

How do I show that there exists inverses, an identity element and that it is closed? I tried first for identity:

WTS there exists e such that ae=a=ea. Then I don't know where to go from there. It seems like I am just assuming that there exists e such that aa'=a(a-inverse)=e

Then from there I can show that there exists an inverse right?

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# Homework Help: Prove that this is a Group

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