Proof that the adjoint representation is an endomorphism

[atat1tata]

Homework Statement

My textbooks takes for granted that, given a Lie group ##g## and its algebra ##\mathfrak{g}##, we have that ##AXA^{-1} \in \mathfrak{g}##.

Homework Equations

For ##Y## to be in ##\mathfrak{g}## means that ##e^{tY} \in G## for each ##t \in \mathbf{R}##

The Attempt at a Solution

I tried to expand ##\exp(tAXA^{-1})## but I am stuck, since the term inside the exponential is a mix of group and algebra matrices and I don't know how to deal with it.

 

[Orodruin]

What can you say in general for a function ##f(AXA^{-1})## assuming it can be Taylor expanded?

Edit: Or rather, what is ##(A X A^{-1})^n##?

 

[atat1tata]

What can you say in general for a function ##f(AXA^{-1})## assuming it can be Taylor expanded?

Edit: Or rather, what is ##(A X A^{-1})^n##?

Thanks! So:
##\exp (tAXA^{-1}) = 1 + tAXA^{-1} + \frac{t^2}{2}AX^2A^{-1} + ... = A(\exp{tX})A^{-1} \in G##

Why I was stuck, I don't understand... Thank you for the hint!