Prove that this is a Group

1. Oct 14, 2008

fk378

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?

2. Oct 14, 2008

Unassuming

What is aut(B)?

3. Oct 14, 2008

fk378

Aut(B) is the set of all automorphisms of G. It is a subgroup of A(G), the set of all 1-1 and onto mappings of G onto itself.

4. Oct 14, 2008

Hurkyl

Staff Emeritus
With this sort of problem it's usually easiest to simply write down an explicit formula for them. The formula usually isn't too hard to guess, and when it's not obvious, it can usually be determined by solving the appropriate equation.

Again, direct calculation -- this one is usually trivial.

5. Oct 14, 2008

fk378

That's where I am stuck--writing the formula. Is the one I wrote with my question completely off?

6. Oct 14, 2008

fk378

Am I supposed to prove there exists an identity or just that the identity is unique?

7. Oct 15, 2008