Factors in finite groups and well-covered graphs

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.

💡 Research Summary

The paper introduces a combinatorial invariant of subsets in finite groups that mirrors the notion of independence in graphs. For a finite group G and a non‑empty subset A⊆G, a (right) s‑factor B⊆G is defined by two requirements: (i) every element of the product set AB can be expressed uniquely as ab with a∈A, b∈B, and (ii) B is maximal with respect to inclusion among subsets satisfying (i). In a finite group the sizes of the largest and smallest such B’s are denoted |G : A|⁺ and |G : A|⁻, called the upper and lower indices of A. When these two numbers coincide, A is called stable; a group in which every subset is stable is called a stable group.

The central insight is that s‑factors are exactly maximal independent sets in a suitable Cayley graph. Let ∂A = A⁻¹A \ {e}. The left Cayley graph Cay(G, ∂A) has vertex set G and edges {g, sg} for s∈∂A. Lemma 2 proves the equivalence: B is a right s‑factor for A iff B is a maximal independent set in Cay(G, ∂A). Consequently, |G : A|⁺ = α(Cay(G, ∂A)) (the independence number) and |G : A|⁻ = i(Cay(G, ∂A)) (the independent domination number, which equals the size of a smallest maximal independent set).

This correspondence translates the problem of computing subset indices into a well‑studied graph‑theoretic problem. The authors exploit it to obtain explicit values for many groups:

• For cyclic groups ℤₙ with A = {0,1}, ∂A = {±1} and the Cayley graph is the n‑cycle Cₙ. Using known formulas for cycles, they obtain |ℤₙ : A|⁻ = ⌈n/3⌉ and |ℤₙ : A|⁺ = ⌊n/2⌋.
• For dihedral groups Dₙ = ⟨a,b | a² = b² = (ab)ⁿ = e⟩, taking S = ∂{e,a,b} = {a,b,ab,ba}, the Cayley graph is 4‑regular with a Hamiltonian cycle and two additional n‑cycles. Careful analysis yields i(Cay(Dₙ,S)) = ⌈2n/5⌉ and α(Cay(Dₙ,S)) = ⌊2n/3⌋.

General structural results are proved about stability. Lemma 4 shows that for a subgroup H of G, the indices of a Cayley graph on G with a symmetric generating set S scale by the index |G : H|. From this, Corollary 5 follows: any subgroup of a stable group must itself be stable. Lemma 6 provides a powerful obstruction: if G has a normal subgroup H with cyclic quotient G/H of order n ≥ 4 and |H| ≥ 3, then G cannot be stable. The proof constructs a specific subset A that forces the associated Cayley graph to have different independence and domination numbers.

Using these tools, the authors systematically eliminate large families of groups from being stable. Sections 4–7 treat: – a non‑abelian group of order 21 and the unitriangular group UT(3,3); – the elementary abelian groups ℤ₅² and ℤ₃³; – the alternating group A₄ and the semidirect product (C₃×C₃)⋊C₂; – several groups of order 16. In each case a suitable subset A is exhibited, and Lemma 2 together with the graph calculations shows that the upper and lower indices differ.

The culmination is a classification theorem (Section 8) describing all finite stable groups without recourse to computer enumeration. The result can be summarized as follows:

1. All cyclic groups Cₙ are stable, with the exception that only n ∈ {1,2,3,4,5,7} yield equality of the two indices for every subset (larger n already fail by Lemma 3).
2. The Klein four‑group V₄ = C₂×C₂ is stable.
3. Certain small 2‑groups (e.g., D₄, Q₈) satisfy stability, but any 2‑group containing a cyclic quotient of order ≥4 with a non‑trivial kernel is excluded by Lemma 6.
4. No non‑abelian group outside the above narrow families is stable.

Thus the paper provides a complete, computer‑free description of the class of finite groups in which every subset has identical upper and lower s‑factor indices.

Beyond the classification, the work establishes a robust bridge between group theory and graph theory. By interpreting s‑factors as maximal independent sets, one can import a wealth of graph‑theoretic techniques (e.g., vertex‑transitivity, domination theory, known bounds for independence numbers) into the study of subset indices. Conversely, group‑theoretic structure (normal subgroups, quotients, coset decompositions) yields useful reductions for graph parameters, as illustrated by Lemma 4. This dual perspective opens avenues for further research, such as exploring s‑factor analogues in infinite groups, investigating algorithmic aspects of computing indices, or applying the framework to combinatorial designs, coding theory, and cryptographic constructions where independence and domination play a role.

The authors also supply GAP code (Section 9) that automates the computation of ∂A, constructs the associated Cayley graph, and determines its independence and domination numbers. This tool enables rapid testing of stability for arbitrary finite groups presented by generators and relations, making the theoretical results readily applicable in computational investigations.

In summary, the paper achieves three major contributions: (1) it introduces the s‑factor concept and links it precisely to maximal independent sets in Cayley graphs; (2) it uses this link to compute subset indices for a variety of groups and to prove a full classification of finite stable groups without computer assistance; and (3) it provides practical computational tools, thereby enriching both the theoretical and applied landscape of combinatorial group theory.