- #1

- 598

- 0

## Homework Statement

[tex]\phi[/tex]:G-->G'

Let [tex]\phi[/tex] be an isomorphism. Prove that [tex]\phi[/tex] maps the e identity of G to e', the identity of G' and for every a[tex]\in[/tex]G, [tex]\phi[/tex]([tex]a^{-1}[/tex])=[tex]^\phi(a){-1}[/tex].

## Homework Equations

## The Attempt at a Solution

We have an isomorphism, therefore one to one, onto and has a homomorphism.

Phi is one to one therefore [tex]\phi[/tex](x)=[tex]\phi[/tex](y), implying x=y.

Then [tex]\phi[/tex](G)=[tex]\phi[/tex](G') implying e=e'.

Now [tex]\phi[/tex](a*[tex]a^{-1}[/tex])=[tex]\phi[/tex](a)*[tex]\phi[/tex]([tex]a^{-1}[/tex]) is what we want to prove.

Now I get stuck.