Factors in finite groups and well-covered graphs

TL;DR

This paper investigates a combinatorial property of subsets in finite groups, called $s$-factors, and establishes a connection with maximal independent sets in Cayley graphs, leading to classifications of stable groups.

Contribution

It introduces the concept of $s$-factors in finite groups, links them to graph theory, and provides a classification theorem for stable groups without computational assistance.

Findings

$s$-factors correspond to maximal independent sets in Cayley graphs.

Upper and lower indices relate to independence and domination numbers.

Complete classification of stable groups achieved without computer calculations.

Abstract

We study a combinatorial property of subsets in finite groups that is analogous to the notion of independence in graphs. Given a group $G$ and a non-empty subset $A\subset G$, we define a (right) $s$-factor as a subset $B\subset G$ satisfying the following conditions: (i) Every element of $AB$ can be written uniquely as $ab$ with $a\in A$ and $b\in B$. (ii) $B$ is maximal (with respect to inclusion) with this property. For a finite group $G$, the upper and lower indices of $A$ are the sizes of the largest and smallest $s$-factors associated with $A$. A subset is called stable if its upper and lower indices coincide. A group is called stable if all its subsets are stable. We then explore the connection between $s$-factors in groups and maximal independent sets in graphs. Specifically, we show that $s$-factors in $G$ associated with $A$ correspond to maximal independent sets in a Cayley graph Cay($G$, $S$), where $S=A^{-1}A\setminus\{e\}$. Consequently, the upper and lower indices of $A$ are equal to the independence number and the independent domination number of the associated Cayley graph. The concepts of $s$-factors, subset indices in groups, stable subsets, and stable groups (under different names) were introduced by Hooshmand in 2020. Later, Hooshmand and Yousefian-Arani classified stable groups using computer calculations. Using the connection with graphs, we compute the upper and lower indices for various groups and their subsets. Furthermore, we prove a classification theorem describing all stable groups without relying on computer calculations.