# Tagged: centralizer

## Problem 455

Let $G$ be a finite group.
The centralizer of an element $a$ of $G$ is defined to be
$C_G(a)=\{g\in G \mid ga=ag\}.$

A conjugacy class is a set of the form
$\Cl(a)=\{bab^{-1} \mid b\in G\}$ for some $a\in G$.

(a) Prove that the centralizer of an element of $a$ in $G$ is a subgroup of the group $G$.

(b) Prove that the order (the number of elements) of every conjugacy class in $G$ divides the order of the group $G$.

## Problem 109

Let $X$ be a subset of a group $G$. Let $C_G(X)$ be the centralizer subgroup of $X$ in $G$.

For any $g \in G$, show that $gC_G(X)g^{-1}=C_G(gXg^{-1})$.

## Problem 94

Let $H$ be a subgroup of order $2$. Let $N_G(H)$ be the normalizer of $H$ in $G$ and $C_G(H)$ be the centralizer of $H$ in $G$.

(a) Show that $N_G(H)=C_G(H)$.

(b) If $H$ is a normal subgroup of $G$, then show that $H$ is a subgroup of the center $Z(G)$ of $G$.

## Problem 54

Determine all the conjugacy classes of the dihedral group
$D_{8}=\langle r,s \mid r^4=s^2=1, sr=r^{-1}s\rangle$ of order $8$.

## Problem 53

Let $D_8$ be the dihedral group of order $8$.
Using the generators and relations, we have
$D_{8}=\langle r,s \mid r^4=s^2=1, sr=r^{-1}s\rangle.$

(a) Let $A$ be the subgroup of $D_8$ generated by $r$, that is, $A=\{1,r,r^2,r^3\}$.
Prove that the centralizer $C_{D_8}(A)=A$.

(b) Show that the normalizer $N_{D_8}(A)=D_8$.

(c) Show that the center $Z(D_8)=\langle r^2 \rangle=\{1,r^2\}$, the subgroup generated by $r^2$.

## Problem 10

Let $G$ be a group of order $|G|=p^n$ for some $n \in \N$.
(Such a group is called a $p$-group.)

Show that the center $Z(G)$ of the group $G$ is not trivial.