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.
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$G$ 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..
Tsemo Aristide
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