Let G be a finite group. The aim of this paper is to study the number of solutions <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$S\subseteq G$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>S</mi> <mo>⊆</mo> <mi>G</mi> </mrow> </math></EquationSource> </InlineEquation> of the equation <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$\mho ^{\{n\}}(S)=L$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msup> <mo>℧</mo> <mrow> <mo stretchy="false">{</mo> <mi>n</mi> <mo stretchy="false">}</mo> </mrow> </msup> <mrow> <mo stretchy="false">(</mo> <mi>S</mi> <mo stretchy="false">)</mo> </mrow> <mo>=</mo> <mi>L</mi> </mrow> </math></EquationSource> </InlineEquation>, where L is a non-empty subset of G, n is a positive integer and <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$\mho ^{\{n\}}(S)=\{ s^n \ | \ s\in S\}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msup> <mo>℧</mo> <mrow> <mo stretchy="false">{</mo> <mi>n</mi> <mo stretchy="false">}</mo> </mrow> </msup> <mrow> <mo stretchy="false">(</mo> <mi>S</mi> <mo stretchy="false">)</mo> </mrow> <mo>=</mo> <mrow> <mo stretchy="false">{</mo> <msup> <mi>s</mi> <mi>n</mi> </msup> <mspace width="4pt" /> <mo stretchy="false">|</mo> <mspace width="4pt" /> <mi>s</mi> <mo>∈</mo> <mi>S</mi> <mo stretchy="false">}</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation>. 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.

A set theoretic version of equations on groups

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2026年3月30日

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Mihai-Silviu Lazorec

摘要

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