Let $G$ be a finite group such that for any two non-identity elements $a,b$ in $G$ , there is an Automorphism of $G$ sending $a$ to $b$ , then is it true that $G$ is abelian ?
2026-03-28 14:05:27.1774706727
From Automorphism to abelian ness ... in a finite group
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Yes. Suppose $G\neq 1$. Since there exists an element with prime order, and all non-identity elements of $G$ have the same order, $G$ is a $p$-group. Therefore $Z(G)\neq 1$. But automorphisms preserve the center, so $Z(G)=G$.