A set theoretic version of equations on groups

Abstract

Let $G$ be a finite group. The aim of this paper is to study the number of solutions $S\subseteq G$ of the equation ${\mho }^{\{n\}}(S)=L$, where $L$ is a non-empty subset of $G$, $n$ is a positive integer and ${\mho }^{\{n\}}(S)=\{s^n|s\in S\}$. Besides our findings obtained in this general frame, we also outline some results which hold for some particular cases such as: \textit{i)} $L$ is a normal subset of $G$; \textit{ii)} $G$ is abelian; \textit{iii)} $G$ is an extraspecial $p$-group.

accepted for publication in Beiträge zur Algebra und Geometrie; 9 pages