Question about Group notation Say I have elements $g$ and $h$ in a group $G$.
What does $g^h$ mean? Seeing this notation a lot but I can't find an explanation for it anywhere.
 A: Usually, it means $h^{-1}gh$. That is, the application of the automorphism $\phi_h:G\to G$ which takes $g\to h^{-1}gh$.
A: In work by group theorists, this is the right action of $G$ on itself by conjugation: $$g^h = h^{-1} g h$$
This has the nice property that $$(gh)^k = g^k h^k \quad \text{and} \quad g^{(hk)} = (g^h)^k$$
The commutator associated with this is $[g,h] = g^{-1} g^h$, the difference between ${}^h$ and ${}^1$, the identity.
You will occasionally see other people use $g^h$ to mean $h gh^{-1}$ as a left-action. Sometimes this is called the topologist's convention, though we have some hope they will all adopt ${}^h g = h g h^{-1}$ so that their left action is on the left.
A: Let $X,Y$ be $G$-sets. Given a space $Y^X$ of maps $f: X \to Y$, the actions on $X$ and $Y$ induce an action on $Y^X$ given by
$$g \cdot f(x) = gf(g^{-1}x)$$
where the $g\cdot f$ on the left hand side indicates the action on $Y^X$ and the $gf$ on the right indicates the action on $Y$. Obviously this notation is very ambiguous, so commonly we write $^g f$ for the action on $Y^X$. The symmetrical right action is generally denoted by an exponent on the opposite side.
Viewing a group element as a function $G \to G$ gives the etymology, so to speak, of this notation.
