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Abstract Algebra

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Group Automorphism: Structure of Automorphism Group of $Z/mZ$

ABSALG-KUCSFP

Let $\mathbb{Z}/m\mathbb{Z}$ be the additive group of integers mod $m$, for $m\ge 1$. The notation $\mathbb{Z}_m$ is also common for this group.

For a group $G$, let ${\rm Aut}(G)$ be the group of bijective group homomorphisms from $G$ to $G$ under composition $\circ$, namely,
for $\psi_1$, $\psi_2$ in ${\rm Aut}(G)$, the automorphism $(\psi_1\circ\psi_2)$ satisfies $(\psi_1\circ\psi_2)(g)=\psi_1(\psi_2(g))$, for all $g\in G$.

Which of the following are correct?

Select ALL that apply.

A

${\rm Aut}(\mathbb{Z}/5\mathbb{Z})$ is a cyclic group of order $4$.

B

${\rm Aut}(\mathbb{Z}/12\mathbb{Z})$ is isomorphic to a direct product of two cyclic groups of order $2$.

C

${\rm Aut}(\mathbb{Z}/7\mathbb{Z})$ is a cyclic group of order $7$.

D

${\rm Aut}(\mathbb{Z}/8\mathbb{Z})$ is isomorphic to a cyclic group of order $4$.

E

${\rm Aut}(\mathbb{Z}/24\mathbb{Z})$ is isomorphic to a direct product of a cyclic group of order $2$ and a cyclic group of order $4$.