The Order of a Conjugacy Class Divides the Order of the Group
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 $G$.
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