1. ## Subgroups, isomorphic

Let H be a subgroup of G and let a be in G. Show that aHa^-1 is a subgroup of G that is isomorphic to H.
I want to show aHa^-1 is 1-1, onto and has a homomorphism property.
Ok so my problem with this is that with isomorphisms before I generally had a function that I worked from, so it was easier to show 1-1, onto, and c(ab)=c(a)c(b), but without a funcction, I get stuck beginning.

2. ## Isomorphism

Let $\phi:H\longrightarrow aHa^{-1}$ by $\phi(x)=axa^{-1}$
Clearly onto and well defined.
To check injectivity, suppose $ax_1a^{-1}=ax_2a^{-1}$ then simply multiply on the left by a and on the right by $a^{-1}$ to conclude $x_1=x_2$
we need only check it is a homomorphism

$\phi(x)\phi(y)=axa^{-1}aya^{-1}=axya^{-1}=\phi(xy)$ proving the isomorphism