Aenea Marin: On finite groups with operators, 505-513

Abstract:

Let $G$ be a finite group with $\vert G\vert\geq 3$ and let $A$ be a non-trivial subgroup of $Aut(G)$. The number $s(g)=\vert\{ (x, \alpha)\in G\times A\ \vert\ g=[x, \alpha]\}\vert$ is determined for an arbitrary $g\in G$. The key to do this is an unpublished observation made by I. M. Isaacs in a particular case.

If $C$ is the set of all commutators $[x, \alpha]$ with $x\in G$ and $\alpha \in A$ and $F$ is the subgroup of the fixed points of $A$ in $G$, we obtain a general and sharp lower bound for $\vert F\cap C\vert$. This is used to define a particular type of action of $A$ on $G$, called here uniform action.

Key Words: Commutators, fixed points, actions, orbits.

2020 Mathematics Subject Classification: Primary 20D45, 20D60, 20F12.

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