A set theoretic version of equations on groups

TL;DR

This paper investigates the number of subsets within finite groups that satisfy specific power equations, extending classical group equation solutions to a set-theoretic context.

Contribution

It introduces a set-theoretic approach to counting solutions of power equations in finite groups, including special cases like abelian and extraspecial p-groups.

Findings

Derived formulas for the number of solutions in general finite groups.

Results specific to normal subsets, abelian groups, and extraspecial p-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.