# Conjugation / Change of basis in groups

If A is a diagonalizable matrix, then $\exists$ P,D such that $P^{-1}AP=D$. This can be viewed as an inner automorphism of $GL(V)$. More generally, I guess one can write that if $\psi:G \times X \to X$ is a group action, and $u \in G$ is a "change of basis" (ie $x_1 = \psi(u,x_0)$ represents $x_0$ after the change of basis), then we have a similar concept by considering the inner automorphism $g'=u^{-1}gu$, $g \in G$, with $g'$ acting on $x_{1}$ or $g$ acting on $x_{0}$.

My question is : what properties or interesting results can one find for general groups, by analogy with the case of matrix groups, and in particular in geometric group theory? The reason of this question is that I'm interested, concerning this group, about all elements of the form $g^{n}wg^{-n}$, where w is a word in the group.

If you consider conjugation in permutation groups, then it does correspond to a sort of change of basis in the sense that $\tau \sigma \tau^{-1}$ is structurally the "same" permutation as $\sigma$ except that you have renamed the elements using $\tau$. In particular, since $S_n$ is generated by $(12)$ and $(123\ldots n)$, it is generated by any $n$-cycle and any transposition of adjacent elements in that $n$-cycle.