Isomorphisms abelian help

1. Aug 20, 2014

Justabeginner

1. The problem statement, all variables and given/known data
Let G be a group, A = G * G. In A, Let T = {(g, g)|g ε G}. Prove that T is isomorphic to G.

2. Relevant equations

3. The attempt at a solution
A is abelian. Therefore, G * G is abelian. T is a subgroup of G.

I am not sure if my above inferences are even correct. Can someone guide me as to the thought process on this please? Thank you.

2. Aug 21, 2014

I don't see how you could have inferred any of those, or why you would need to.

You are asked to show that $G$ and $T$ are isomorphic. There is an obvious candidate for an isomorphism, so you should just verify that it actually is one.

3. Aug 21, 2014

LCKurtz

And for that matter, you haven't told us what G*G means.

4. Aug 22, 2014

Justabeginner

A = G * G, as in G cross G.
I do not understand how T is directly isomorphic to G.

5. Aug 22, 2014

vela

Staff Emeritus
How'd you get A is abelian?

If A is abelian, then obviously GxG is abelian since A=GxG.

How can T be a subgroup of G when it's not a subset of G? It's a subset of A, right?

How did you define the group multiplication for A?

6. Aug 23, 2014

pasmith

There is nothing in the question to suggest this. If $G$ is not abelian then $A$ will not be abelian.

Actually T is a subgroup of A.

You need to find a bijection $\phi : G \to T$ such that $\phi(g)\phi(h) = \phi(gh)$ for every $g \in G$ and $h \in G$.

It would be good to start by writing out the group operation of A, and see what happens when you restrict it to T.

7. Aug 23, 2014

LCKurtz

What is the group operation on G*G? You have to know that before you can even talk about an isomorphism.