?

Abstract Algebra

Free Version

Upgrade subject to access all content

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?

A

$C({\rm GL}(2,{\mathbb R}))$ consists of all the invertible diagonal matrices

B

$C_{{\rm GL}(3,{\mathbb R})}(D)$, where $D$ is a $3\times 3$ invertible diagonal matrix, consists only of the $\lambda I_3$, $\lambda\not=0$

C

In ${\rm GL}(2,{\mathbb R})$, all upper triangular matrices commute with all lower triangular matrices

D

$C_{{\rm GL}(2,{\mathbb R})}({\rm SL}(2,{\mathbb R}))$ consist of all the invertible diagonal matrices

E

$C({\rm SL}(2,{\mathbb R}))$ consists only of $I_2$ and $-I_2$