Moderate# Center and Centralizer of a Group: definition and example

ABSALG-RIQC4X

If $g,h\in G$ satisfy $gh=hg$, we say **equivalently** that $g$ commutes with $h$, **or** $h$ commutes with $g$, **or** $g$ and $h$ commute.

The center $C(G)$ of $G$ (also denoted $Z(G)$) is given by:

$\{g\in G\mid gh=hg\;{\rm for\, all}\; h\in G\}$

Let $S$ be a subset of $G$, and let $H$ be a subgroup of $G$.

The centralizer $C_H(S)$ of $S$ in $H$ is given by:

$\{h\in H\mid hs=sh\;{\rm for\, all}\;s\in S\}$

We write $C_H(s)$ for $C_H(\{s\})$, $s\in G$. Notice that $C_G(G)=C(G)$.

For $n\ge 1$, denote by:

${\rm GL}(n,{\mathbb R})$ the group of $n\times n$ matrices with non-zero determinant

${\rm SL}(n,{\mathbb R})$ the group of $n\times n$ matrices with determinant equal $1$

$I_n$ the $n\times n$ identity matrix

Which of the following is **TRUE**?