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.

Solution 1:

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.

Solution 2:

Usually, it means $h^{-1}gh$. That is, the application of the automorphism $\phi_h:G\to G$ which takes $g\to h^{-1}gh$.