Must the centralizer of an element of a group be abelian?
I see that the definition of centralizer is:
Let $a$ be a fixed element of a group $G$. The centralizer of $a$ in $G$, $C(a)$, is the set of all elements in $G$ that will commute with $a$. In symbols, $C(a)=\{g\in G \mid ga=ag\}$.
But this doesn't necessarily mean that the centralizer is abelian, does it?
Best Answer
Let $G$ be any non-abelian group, and let $e$ be the identity of the group. For all $g\in G$, we have $$ge=eg=g,$$ so $C(e)=G$ is non-abelian.