Moderate# Groups Automorphisms: Conjugation Automorphism of a Group

ABSALG-VAR1LG

Let $G$ be a group and let $g\in G$. The automorphism of $G$ given by:

$h\mapsto ghg^{-1}$, for all $h\in G$

...is called **conjugation by $g$** and $h$ and $ghg^{-1}$ are said to be conjugate.

The set ${\bar h}$ of all elements in $G$ conjugate to $h$ is called the conjugacy class of $h$.

If $H$ is a subgroup of $G$, the subgroup of $G$ given by $gHg^{-1}$ is said to be **conjugate to $H$**.

For $g\in G$, let $C_G(g)$ be the centralizer of $\{g\}$ in $G$ and,

for a subgroup $H$ of $G$, let $N_G(H)$ be the normalizer of $H$ in $G$.

Now, let $G$ be a finite group, and let $\langle h\rangle$ be the cyclic subgroup of $G$ generated by $h\in G$.

Which of the following are **always true?**

Select **ALL** that apply.