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Abstract Algebra

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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.

A

${\bar h}$ has cardinality $[G:N_G(\langle h\rangle)]$

B

The number of subgroups conjugate to $\langle h\rangle$ is $[G: ​C_G(h)]$

C

If $G$ is the disjoint union of ${\bar h}_i$, $h_i\in G$, $i=1,\ldots,k$, then:

$|G|=\sum_{i=1}^k[G:C_G(h_i)]$

D

The map $A:G\rightarrow {\rm Aut}(G)$, associating to $g\in G$ the automorphism

conjugation by $g$ ​is an injective group homomorphism.

E

For $g\in G$, let $\sigma_g:G\rightarrow G$ be given by $\sigma_g(h)=ghg^{-1}$.

The map $g\mapsto \sigma_g$ is a group homomorphism from

$G$ to the group of permutations of the set $G$.