Proof of a linear operator acting on an inverse of a group element

[Dixanadu]

Hey guys!

Basically, I was wondering how to prove the following statement. I've seen it in the Hamermesh textbook without proof, so I wanted to know how you go about doing it.

Let's say you have a group element $g_{1}$, which has a corresponding inverse $g_{1}^{-1}$. Let's also define a linear transformation D for this group element.

So what I am trying to prove is that

$D(g_{1}^{-1}) = [D(g_{1})]^{-1}$

Can u guys point me in the right direction?

Thanks!

 

[Office_Shredder]

By linear transformation do you mean group homomorphism? If so just consider
$$D(g_1 g_1^{-1}) = D(g_1) D(g_1^{-1})$$

 

[Dixanadu]

yes it is a homomorphism. But what u wrote in the previous message, where do I go from there? cos $D(g_{1}g_{1}^{-1})$ is just $D(E)$ where E is the identity, right? o.o

 

[Office_Shredder]

Yes, and what is D(E) equal to?

 

[Dixanadu]

OHH I get it. Basically, because $D(E) = E$, we can say that

$E = D(g_{1})D(g_{1}^{-1})$
Then by multiplying both sides on the left by $[D(g_{1})]^{-1}$ we get

$[D(g_{1})]^{-1} = D(g_{1}^{-1})$...I think...