Aut-Z-where-Aut-G-f-f-G-G-is-a-group-f-is-a-isomorphism-G-

Question Number 204018 by mnjuly1970 last updated on 04/Feb/24

A u t (Z) = ?

w h e r e, A u t (G) = {f ∣ f : G \underset{G i s a g r o u p}{\overset{f i s a i s o m o r p h i s m}{\to}} G}

Answered by witcher3 last updated on 04/Feb/24

f \in Aut (G)

f (0) = 0

let f (1) = a \in Z

\Rightarrow f (n) = na

suppose ∣ a ∣> 1 \Rightarrow \exists p \in IP p ∣ a

\Rightarrow 1 \notin f (Z) ? since 1 = ap \Rightarrow p ∣ 1 impossibl

\Rightarrow∣ a ∣⩽ 1

a = 0 \Rightarrow f = 0 not bijective

\Rightarrow a \in {1, - 1}

f (x) = {\begin{cases} x \\ - x \end{cases}

f = id; - id worcks

Commented by mnjuly1970 last updated on 04/Feb/24

Commented by witcher3 last updated on 04/Feb/24

withe Pleasur