A topological group and inner invariant group

Question

A topological group G is called inner invariant group if there is a compact neighborhood $U$ of $e$ with ‎$‎ xUx‎^{-1} ‎\subseteq ‎U‎$ ‎for ‎‎$‎x\in G‎$‎. show that discrete groups, compact group, and abelian group are inner invariant group.

Answer 1

$G$ is discrete if and only if there exists a neighborhood $U$ of $e$ such that $U=\{e\}$, so for every $x\in G$, $xUx^{-1}=\{e\}=U$.

$G$ is compact, take $U=G$.

$G$ is Abelian and locally compact, take $U$ any compact neighborhood of $e$, $xUx^{-1}$ is $U$ since $Ad(x)$ defined by $Ad(x)(y)=xyx^{-1}=x$ is the identity..