Understanding the non-trivial automorphism which exists on $G$

Question

Assume that $G$ is a finite group such that $|G| \gt 1$ and there exists $x \in G$ such that $x^2 \neq e$. Prove that there exists a non-trivial automorphism on $G$.

Note: This question has been asked and answered here. But, I can't understand that answers. Both of them use the concept of inner-automorphism. We haven't studied this in the class, So i'm not allowed to use the term 'Inner-Automorphism'. Also, I don't understand that what is their automorphism doing. We are given a group. Maybe its abelian. Maybe it's not. We should provide a function which takes an element of $G$ and maps it to another element of $G$. Which element is that another element?

You will need to know what an inner automorphism is of $G$. – Dietrich Burde May 12 '17 at 14:03 — Dietrich Burde, May 12 '17 at 14:03

lhf · Accepted Answer · 2017-05-12T14:24:20.953

4

If $G$ is abelian, then $\phi:g \mapsto g^{-1}$ is a non-trivial automorphism because $\phi(x)\ne x$.

If $G$ is not abelian, there are elements $a,b \in G$ such that $ab\ne ba$. Then $\phi:g \mapsto a^{-1}ga$ is a non-trivial automorphism because $\phi(b)\ne b$.

Thus, the existence of $x \in G$ such that $x^2 \neq e$ in only used in the abelian case, but as the original answer notes, it is not really needed.

edited May 12 '17 at 14:24

answered May 12 '17 at 14:02

lhf

221,500

@llrf If G is not abelian , in your example just take $x \notin Z(G)$. – Riju May 12 '17 at 14:06
This answer is just a distillation of https://math.stackexchange.com/a/8401/589. – lhf May 12 '17 at 14:12
This may be a distillation... But i can understand it :) That's the point :) and i appreciate that :) – Arman Malekzadeh May 12 '17 at 14:26

Understanding the non-trivial automorphism which exists on $G$

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