Group generation and normal subgroups

[Bleys]

There is a step in some book which is vaguely explained, I just want to check whether my working is correct.
Let G be a group, and let x be in G-{1}. Let Y be the set $\left\{g^{-1}xg : g\in G\right\}$.
I want to show <Y> is normal in G. Now it's clear why Y is invariant under conjugation (Y being itself a conjugacy class, namely of x). The book says that from this it follows <Y> is normal, but I didn't really think it that was clear. Why does it follow that also <Y> is invariant under conjugation?

This is what I've done, but I don't know if it's correct (or if it's even the argument the book is implying):
any y in <Y> is of the form $g^{-1}x^{k}g$ where k is a positive integer, or $gx^{k}g^{-1}$ where k is a negative integer. I'll do k=2;the rest follows by induction.
First case: $g^{-1}g^{-1}x^{2}gg = g^{-1}(g^{-1}xg)(g^{-1}xg)g = g^{-1}(g^{-1}xg)gg^{-1}(g^{-1}xg)gg$Since Y is invariant under conjugation, this is a product of two elements in Y hence in <Y>.
Second case: $g^{-1}gx^{-2}g^{-1}g = x^{-2}$, which is clearly in <Y> since x is in Y.
Then for any y in <Y>, $g^{-1}yg$ is in <Y>.
Is this correct? Is there a simpler way to do it?

EDIT: I don't know why the code is not showing for the First case...

 

[Tinyboss]

It's because $$g^{-1}(ab)g=(g^{-1}ag)(g^{-1}bg)$$, i.e. conjugation distributes over products, so you only need to verify invariance under conjugation for a set of generators.

 

[Bleys]

oh ok; thanks Tinyboss!