Subgroup Perfect Codes in Cayley Graphs

• Subgroup perfect codes are subgroups of finite groups that serve as perfect codes in Cayley graphs, ensuring every vertex outside the subgroup is uniquely adjacent to one subgroup element.
• The local complement criterion and Diamond Lemma provide a structural framework to determine when a subgroup can function as an efficient dominating set.
• Methods adapted from circulant graph domination are used to classify maximal subgroup perfect codes, offering actionable insights for both theoretical study and algorithmic detection.

A subgroup perfect code is a subgroup of a finite group that serves as a perfect code (efficient dominating set) in some Cayley graph of the group. The paper of subgroup perfect codes in Cayley graphs connects group theory, combinatorics, and the structural theory of error-correcting codes. Recent research has emphasized necessary and sufficient conditions for maximal subgroups of a finite group to act as perfect codes, structural reduction techniques, and methods for explicit construction or classification across broad group families.

1. Formal Framework and Definitions

A perfect code in a graph $\Gamma = (V, E)$ is a set $C \subseteq V$ such that every vertex in $V \setminus C$ is adjacent to exactly one vertex in $C$, and no two vertices in $C$ are adjacent. In the setting of group theory, if $\Gamma$ is the Cayley graph $\mathrm{Cay}(G, S)$ of a finite group $G$ with inverse-closed connection set $S \subseteq G \setminus \{1\}$, a subgroup $H$ of $G$ is a subgroup perfect code if there exists $S$ making $H$ a perfect code in $\mathrm{Cay}(G, S)$. The emphasis on maximal subgroups $M < G$ interrogates which group–subgroup pairs admit this combinatorial tiling structure (Qiao et al., 31 Jul 2025).

The essential algebraic–combinatorial link is the existence of an inverse-closed left transversal $T \subseteq G$ for $H$, ensuring every group element can be written uniquely as $th$ with $t \in T$, $h \in H$. This is equivalent, in most group types, to finding “local complements” for $H$ within certain overgroups and to the presence of specific semidirect product decompositions detailed below (Chen et al., 2019, Zhang et al., 2020).

2. Local Complementation and the Main Criterion

A central structural result characterizes when a $2$-subgroup $H$ of a finite group $G$ is a perfect code via the local structure in the normalizer $N_G(H)$ (Qiao et al., 31 Jul 2025). The necessary and sufficient condition is as follows:

Local Complement Criterion:

Let $H$ be a $2$-subgroup of $G$. Then $H$ is a perfect code of $G$ if and only if for every $a \in N_G(H) \setminus H$ with $a^2 \in H$, the subgroup $H \langle a \rangle$ splits as the semidirect product $H \rtimes \langle a \rangle$—that is, $H$ has a complement of order $2$ in $H \langle a \rangle$. This guarantees an inverse-closed left transversal exists.

This criterion generalizes to arbitrary maximal subgroups $M$ when the Sylow $2$-subgroup case is used as a reduction (Qiao et al., 31 Jul 2025, Zhang, 2022). The semidirect product requirement encodes the capacity to “resolve” possible obstructions arising from involutive elements outside $H$ whose squares land inside $H$.

3. The Diamond Lemma and Reduction Techniques

A principal structural tool is the "Diamond Lemma" (Qiao et al., 31 Jul 2025), a two-layer result analogous to lattice theory transitivity, formalized as follows. Suppose $G$ has subgroups $H, K$ such that $H \cap K$ is a perfect code in $K$. Then $H$ is a perfect code in $HK$. If the $2$-parts of $H$ and $H \cap K$ coincide, the converse holds.

This lemma enables the "lifting" of perfect code properties from subgroups and quotient groups, thereby constructing maximal subgroup perfect codes from corresponding subgroups in normal sections or index subgroups. In particular, Theorem 4.5 in (Qiao et al., 31 Jul 2025) shows that for any group extension $G = K \rtimes Q$, a maximal subgroup $M$ with $K \leq M$ is a perfect code in $G$ if and only if its image in $G/K$ is a perfect code.

Such reduction strategies are instrumental in classifying perfect codes among maximal subgroups in the O'Nan–Scott types: holomorphs of abelian groups (HA), HS, HC, TW, SD, CD, and almost simple (AS) types (Qiao et al., 31 Jul 2025).

4. Primitive and Almost Simple Groups: Classification of Maximal Subgroup Perfect Codes

For several primitive group types, every point-stabilizer is proved to be a perfect code via the existence of regular or transitive normal subgroups and the Diamond Lemma (Qiao et al., 31 Jul 2025). In almost simple groups (particularly those with socle $PSL(2,q)$ or $A_n$), a more nuanced analysis is essential.

For $PSL(2,q)$, the main exceptions occur for large $q$ in the dihedral-type maximal subgroups:

• For $q > 7$, $q \equiv -1 \pmod{8}$ and $M \cong D_{q-1}$, or
• For $q > 9$, $q \equiv 1 \pmod{8}$ and $M \cong D_{q+1}$, $M$ is not a perfect code (Qiao et al., 31 Jul 2025). Further exceptional cases arise due to the action of certain outer automorphisms, as fully classified in Theorem 6.5 of (Qiao et al., 31 Jul 2025). For groups of type $AS$ with socle $A_n$, techniques using wreath products are employed; in many cases, perfect code status is preserved under wreath product extensions, but the converse can fail (Corollary 7.9).

5. Connections to Circulant Graphs, Efficient Domination, and Broader Context

Earlier work on efficient domination (perfect codes) in circulant graphs (Cayley graphs of cyclic groups) with two chord lengths [Peters, Ruži, 2007] and in "metric-induced" configurations [Medina, Beivide, Gabidulin] laid the foundation for graph-theoretic approaches to perfect codes. These studies focus on explicit enumeration and construction in highly symmetric graphs.

Techniques from circulant graph domination theory (inverse-closed transversals, local complementations, two-local analysis) are parallel to those employed for arbitrary Cayley graphs in (Qiao et al., 31 Jul 2025), and are referenced for context and methodological continuity. For example, in circulant graphs, the efficient dominating set problem corresponds to subgroup perfect codes in cyclic settings, fully described via tiling by coset representatives (Ma et al., 2019, Huang et al., 2016).

6. Table: Characterization Criteria for Maximal Subgroup Perfect Codes

Group Situation	Key Criterion	References
2-subgroup $H \leq G$	$H$ has order-2 complement in $H\langle a\rangle$ for all $a \in N_G(H)\setminus H$	(Qiao et al., 31 Jul 2025)
Maximal $M$ with $K\triangleleft G,\ K \leq M$	$M/K$ is perfect code in $G/K$	(Qiao et al., 31 Jul 2025)
$PSL(2,q)$, maximal $M$	Exceptions for $D_{q-1}, D_{q+1}$ as above; otherwise $M$ is a perfect code	(Qiao et al., 31 Jul 2025)
Primitive (HA, HS, HC, TW, SD, CD) types	Every stabilizer perfect code via Diamond Lemma	(Qiao et al., 31 Jul 2025)

7. Implications and Open Questions

The characterization of maximal subgroup perfect codes via local complement conditions unifies structural, algebraic, and combinatorial perspectives. The result that transversality or tiling with involutive resolutions is necessary and sufficient for efficient domination generalizes both previous results on abelian and dihedral cases and recent reduction techniques to $2$-groups (Zhang, 2022, Ma et al., 2019).

Further directions include:

• Systematic exploration of perfect code existence in additional almost simple group types.
• Analysis of the possible failure of the converse implications in wreath products and other group extensions.
• Extension and adaptation of techniques from circulant graph domination to broader non-cyclic, non-abelian settings.
• Algorithmic detection of local complements and efficient dominating sets in large or computationally constructed groups.

The theoretical advances bridge algebraic coding theory, group analysis, and combinatorics, showing that maximal subgroup perfect codes reflect subtle local group symmetries and have direct, rigorously characterizable graph-theoretic consequences.