Let $G$ be a finite group with at least four elements such that $g^2=e$ for all $g \in G$. Show that there is a non-trivial automorphism on $G$.
How do I proceed to prove it? Please help me in this regard.
Thank you very much.
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little o
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If $G$ has more than $2$ elements, then there exists a non-trivial automorphism. This is satisfied here. Arturo's proof is quite short, and refers to elements of order $2$. – Dietrich Burde Aug 15 '18 at 18:17
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I don't understand why every group whose every element is of exponent $2$ is a vector space over a field of $2$ elements. Where is the property of the group used? If some element of $G$ is not of index $2$ then what are the problems? Please help me in this regard. – little o Aug 15 '18 at 18:31
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Dear Debabrata, see this question or more generally this question for a good explanation (by Jyrki, take $p=2$). – Dietrich Burde Aug 16 '18 at 08:51