What question did this study set out to answer?

This paper aims to explore the number of solutions to a specific equation defined on finite groups.

April 1, 2026Open Access

A set theoretic version of equations on groups

Key Points

This paper aims to explore the number of solutions to a specific equation defined on finite groups.
Studied finite groups G with respect to solutions S that satisfy $$℧^{\{n\}}(S)=L$$.
Considered special cases such as normal subsets, abelian groups, and extraspecial p-groups.
Utilized set theoretic approaches to analyze the structure and solutions.
Identified conditions under which the equation has solutions.
Outlined distinct findings for normal subsets and specific group types.
Provided insights into the behavior of the solutions based on group properties.

Abstract

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

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