Factors in finite groups and well-covered graphs

TL;DR

The paper establishes a tight bridge between right $s$-factors in finite groups and maximal independent sets in Cayley graphs via $\partial A= A^{-1}A\setminus\{e\}$, showing $|G:A|^+=\alpha(\mathrm{Cay}(G,\partial A))$ and $|G:A|^-=i(\mathrm{Cay}(G,\partial A))$. This graph-theoretic lens yields concrete computations of indices for a wide range of groups (including dihedral, specific order-21 and order-27 groups, $\mathbb{Z}_2^5$, $\mathbb{Z}_3^3$, $A_4$, and certain order-16 groups) and culminates in a complete classification: only 14 finite groups are stable, with detailed argument avoiding computer enumeration. A key methodological contribution is reducing questions about stability to well-covered properties of Cayley graphs, supported by explicit constructions and a GAP verification toolkit. The results illuminate the structure of stable groups and connect independent domination and independence notions in Cayley graphs to subset indices in groups, with potential impact on combinatorial group theory and graph theory.

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.

Figures (6)

• Figure 1: For Lemma \ref{['lemma:On cyclic quotient group']}. The set $I=\{v_0,\dots,v_k\}$ lies in the even cosets. For each $i$, $N[v_i]=(A^{-1}A)v_i$ covers $Hg^{2i}$ and covers each adjacent odd coset $Hg^{2i\pm1}$ except for two “holes”: $\eta v_i$ and $\eta'v_i$. The condition $\eta v_i\neq \eta'v_{i+1}$ ensures $Hg^{2i+1}\subset N[v_i]\cup N[v_{i+1}]$. In the odd case, $v_k=g^{-1}h\in Hg^{n-1}$ and the unique non-neighbor of $v_k$ in $Hg^{2k-1}$ is $(g^{-1}h)^2$.
• Figure 2: Graphs $\mathop{\mathrm{Cay}}\nolimits(D_4,S)$, $\mathop{\mathrm{Cay}}\nolimits(D_5,S)$, $\mathop{\mathrm{Cay}}\nolimits(D_6,S)$
• Figure 3: The subgraph of $\mathop{\mathrm{Cay}}\nolimits(G,S)$ with vertex set $G\setminus N[e]$, $|G|=21$
• Figure 4: The subgraph of $\mathop{\mathrm{Cay}}\nolimits(G,S)$ with vertex set $W=G\setminus N[e]$, where $|G|=27$
• Figure 5: The Cayley graph $\mathop{\mathrm{Cay}}\nolimits(A_4,\partial A)$ is the icosahedral graph

Theorems & Definitions (32)

• Lemma 1
• proof
• Lemma 2
• proof
• Lemma 3: see HooshmandArani, Theorem 4.1
• proof
• Lemma 4
• proof
• Corollary 5
• Lemma 6