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1989•Proceedings of the American Mathematical SocietyOpen access

On Inner Automorphisms of Finite Groups

Martin R. Pettet

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Abstract

It is shown that in certain classes of finite groups, inner automorphisms are characterized by an extension property and also by a dual lifting property. This is a consequence of the fact that for any finite group $G$ and any prime $p$, there is a $p$-group $P$ and a semidirect product $H = GP$ such that $P$ is characteristic in $H$ and every automorphism of $H$ induces an inner automorphism on $H/P$.

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What this paper is about

It is shown that in certain classes of finite groups, inner automorphisms are characterized by an extension property and also by a dual lifting property. This is a consequence of the fact that for any finite group $G$ and any prime $p$, there is a $p$-group $P$ and a semidirect product $H = GP$ such that $P$ is characteristic in $H$ and every automorphism of $H$ induces an inner automorphism on $H/P$.

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Available abstract

It is shown that in certain classes of finite groups, inner automorphisms are characterized by an extension property and also by a dual lifting property. This is a consequence of the fact that for any finite group $G$ and any prime $p$, there is a $p$-group $P$ and a semidirect product $H = GP$ such that $P$ is characteristic in $H$ and every automorphism of $H$ induces an inner automorphism on $H/P$.

Key concepts: Automorphism, Semidirect product, Outer automorphism group, Mathematics, p-group, Inner automorphism, Pure mathematics, Prime (order theory)

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