# 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.