?

Abstract Algebra

Free Version

Upgrade subject to access all content

Easy

Group Automorphisms: symmetric and cyclic groups

ABSALG-XNOA1V

Let $G$ be a group. An automorphism of $G$ is a group isomorphism $f:G\rightarrow G$.

The set ${\rm Aut}(G)$ of all automorphisms of $G$ is a group under composition of maps.

Let $S_n$ be the symmetric group on $n\ge 1$ objects and let $C_n$ be any cyclic group of order $n\ge 1$

Which of the following are TRUE?

Select ALL that apply.

A

The map $g\rightarrow g^{-1}$ is an automorphism of $C_6$

B

The map $g\rightarrow g^{-1}$ is an automorphism of $S_3$

C

The group ${\rm Aut}(C_6)$ is isomorphic to $C_6$

D

The group ${\rm Aut}(C_2\times C_2)$ is isomorphic to $C_2$

E

The group ${\rm Aut}(S_3)$ is isomorphic to $S_3$