I encountered the notion of a topological group and a continuous group action recently and have a basic question about a possible alternative characterization and whether it captures the same notion or not. I'm curious whether a group $G$ (where $^{-1}$ is continuous) acting continuously on itself by conjugation is sufficient to be a topological group.
Is there a further non-topological property that we could impose on our group $G$ so that $G$ acting on itself continuously by conjugation would be equivalent to $G$ being a topological group. For example, we know that $G$ cannot be Abelian.
Suppose I have a group $G = (G_0, ^{-1}, *)$ equipped with a topology $\tau$. $G$ is a topological group if and only if the inverse map $^{-1} : G_0 \to G_0$ is continuous and $G$ acts continuously on itself by left multiplication (or equivalently right multiplication).
However, in addition to acting on itself by left or right multiplication, every group acts on itself by conjugation.
If I know that $^{-1} : G_0 \to G_0$ is continuous and $G$ acts on itself by conjugation continuously (i.e. $x \mapsto gxg^{-1}$ is continuous for all $g$), what further constraints do I need to impose so that $G$ is a topological group?