Subgroup Perfect Code: Theory & Applications

• Subgroup perfect codes are subgroups that serve as perfect codes in Cayley graphs, defined by independent sets whose closed neighborhoods partition the vertex set.
• They bridge graph theory and group theory by connecting efficient domination, completely regular codes, and transversal-based tiling methods.
• Recent research refines criteria for both normal and non-normal subgroup perfect codes, emphasizing 2-local structures and quotient-normalizer reductions.

A subgroup perfect code is a subgroup $H\le G$ that appears as a perfect code in some Cayley graph on $G$; equivalently, $H$ is an independent set whose closed neighborhoods partition the vertex set of a suitable graph built from the group. In the contemporary literature the notion has two main forms: the classical Cayley-graph version, where one speaks of a subgroup perfect code of a group $G$, and the coset-graph version, where a subgroup $A$ containing $H$ is a perfect code of a pair $(G,H)$. The topic lies at the intersection of perfect $1$-error-correcting codes, efficient domination, inverse-closed transversals, group tilings, completely regular codes of covering radius $1$, and equitable $2$-partitions (Chen et al., 2019, Abdollahi et al., 24 Dec 2025).

1. Graph-theoretic and coding-theoretic foundations

A perfect code in a graph $G$0 is a subset $G$1 such that $G$2 is independent and every vertex in $G$3 is adjacent to exactly one vertex in $G$4. Equivalently, the closed neighborhoods $G$5 form a partition of $G$6. This is the standard graph-theoretic notion of an efficient dominating set, and in regular graphs it is exactly the $G$7-regular case of the broader theory of $G$8-regular sets (Huang et al., 2016, Abdollahi et al., 24 Dec 2025).

The regular-set viewpoint is structurally important. If $G$9 is regular, then an $H$0-regular set is a subset $H$1 for which every vertex in $H$2 has exactly $H$3 neighbors in $H$4 and every vertex outside $H$5 has exactly $H$6 neighbors in $H$7. In this language, a perfect code is the special case $H$8. The same condition can be read as a completely regular code with covering radius $H$9 and as an equitable $G$0-partition of the vertex set. This suggests that subgroup perfect codes are not merely isolated dominating sets: they are highly constrained regular configurations inside vertex-transitive graphs (Abdollahi et al., 24 Dec 2025).

A related notion is the total perfect code, where every vertex of the graph has exactly one neighbor in the code. Total perfect codes are not subgroup perfect codes in the strict sense, but they recur throughout the theory because many structural criteria are first proved in parallel for perfect and total perfect codes and then specialized to the subgroup case (Huang et al., 2016).

2. Cayley graphs, transversals, and tilings

For a finite group $G$1 and an inverse-closed subset $G$2, the Cayley graph $G$3 has vertex set $G$4 and adjacency determined by $G$5. A subgroup $G$6 is a subgroup perfect code of $G$7 if there exists such an $G$8 for which $G$9 is a perfect code in $A$0. In a fixed Cayley graph, the basic characterization is transversal-theoretic: $A$1 is a perfect code in $A$2 if and only if $A$3 is a left transversal of $A$4 in $A$5; similarly, $A$6 is a total perfect code if and only if $A$7 itself is a left transversal (Huang et al., 2016).

At the group level, the intrinsic formulation is sharper. A subgroup $A$8 is a perfect code of $A$9 if and only if it admits an inverse-closed right transversal in $H$0. Equivalently, for each $H$1 such that $H$2 and $H$3 is odd, there exists $H$4 with $H$5 (Chen et al., 2019). This is one of the central structural characterizations in the subject, because it replaces graph-theoretic adjacency by a condition on cosets, conjugation, and involutions.

The tiling interpretation is equally fundamental. A subgroup perfect code is the same as a factorization $H$6 with unique expression $H$7, where $H$8 is inverse-closed and contains the identity. In group-ring language, perfect-code existence becomes a tiling identity, and for subgroups this makes the perfect-code problem a special case of factorization theory in finite groups (Huang et al., 2016). A plausible implication is that subgroup perfect codes are best understood not as arbitrary dominating sets, but as symmetric coset selectors constrained by inversion.

3. Normal subgroups, non-normal subgroups, and corrected criteria

For normal subgroups, the theory is especially explicit. If $H$9, then $(G,H)$0 is a perfect code of $(G,H)$1 if and only if

$(G,H)$2

The same condition, together with $(G,H)$3 even, characterizes when $(G,H)$4 is a total perfect code. Several immediate consequences follow: if $(G,H)$5 is odd or $(G,H)$6 is odd, then $(G,H)$7 is a perfect code; for cyclic groups this yields a complete order-and-index criterion; and for dihedral groups it leads to an explicit classification of subgroup perfect and subgroup total perfect codes (Huang et al., 2016).

For arbitrary subgroups, Chen–Wang–Xia’s characterization replaces normality by a coset condition. A subgroup $(G,H)$8 is a perfect code if and only if it has an inverse-closed right transversal, or equivalently if every $(G,H)$9 with $1$0 and $1$1 odd yields a coset $1$2 containing an involution (Chen et al., 2019). This formulation is sufficiently flexible to analyze non-normal cases, 2-subgroups, and double-coset obstructions.

A later corrigendum refined part of the general theory. In particular, reduction from $1$3 to the normalizer $1$4 is valid only under an extra parity hypothesis: $1$5 is a $1$6-group, or at least one of $1$7 and $1$8 is odd. The same corrigendum reformulated the obstruction to perfect-code existence in terms of inverse-closed double cosets having an odd number of left cosets and containing no involution (Zhang et al., 2022). This correction matters because some earlier normalizer reductions are false without those hypotheses.

One common overextension is to treat the normal-subgroup square condition as if it remained necessary for non-normal subgroup perfect codes. That is false. An infinite family in $1$9, with $1$0, gives non-normal subgroup perfect codes $1$1 for which there exists $1$2 such that $1$3 but $1$4 for all $1$5. The correct control object in that family is the normalizer $1$6, not the ambient group $1$7 itself (Behajaina et al., 2021).

4. Extension to vertex-transitive graphs and regular sets

Every vertex-transitive graph can be represented as a coset graph $1$8, where the vertex set is the set of left cosets of $1$9 in $2$0, and adjacency is determined by an inverse-closed union of double cosets $2$1. This observation extends subgroup perfect codes from Cayley graphs to arbitrary vertex-transitive graphs: if $2$2, then $2$3 is called a perfect code of the pair $2$4 when the set $2$5 of left cosets of $2$6 in $2$7 is a perfect code in some coset graph $2$8 (Wang et al., 2021, Abdollahi et al., 24 Dec 2025).

In the pair setting, the analogue of the inverse-closed transversal criterion is exact. A subgroup $2$9 with $G$00 is a perfect code of $G$01 if and only if there exists a left transversal $G$02 of $G$03 in $G$04 such that

$G$05

For total perfect codes of $G$06, one needs in addition that the transversal contains an element of $G$07 (Wang et al., 2021). This is the natural generalization of the Cayley-graph criterion, since the case $G$08 recovers the classical inverse-closed transversal condition.

The regular-set generalization goes further. In vertex-transitive graphs, subgroup perfect codes become the $G$09-special case of subgroup $G$10-regular sets. For $G$11, $G$12 is an $G$13-regular set of $G$14 if there exists $G$15 in which $G$16 is $G$17-regular. Theorem A gives a transversal characterization: $G$18 is an $G$19-regular set of $G$20 if and only if there exists $G$21 such that $G$22, $G$23 is a union of exactly $G$24 left cosets of $G$25, and for each $G$26, $G$27 is a union of exactly $G$28 left cosets of $G$29 (Abdollahi et al., 24 Dec 2025).

In the normal case $G$30, the perfect-code problem admits a quotient-normalizer reduction. If $G$31 is normal and contains $G$32, then

$G$33

This quotient-normalizer criterion is one of the strongest recent structural results, because it shows that subgroup perfect codes on the coset level $G$34 are controlled by subgroup perfect codes in the smaller quotient $G$35 (Abdollahi et al., 24 Dec 2025).

Within ordinary Cayley graphs, the regular-set viewpoint has also been pushed to full equivalence. For an arbitrary subgroup $G$36, $G$37 is a perfect code of $G$38 if and only if it is an $G$39-regular set of $G$40 for every pair $G$41, $G$42, with $G$43; moreover, every subgroup is an $G$44-regular set when $G$45 is even (Khaefi et al., 2023). This suggests that, in Cayley graphs, subgroup perfect codes are equivalent to a much stronger universal regularity property.

5. Cayley sum graphs and related subgroup-based graph models

A separate branch of the subject replaces Cayley graphs by Cayley sum graphs. For a finite abelian group $G$46 and a square-free subset $G$47, the Cayley sum graph $G$48 has vertex set $G$49 and adjacency $G$50. In that setting, a subgroup $G$51 is a subgroup perfect code if and only if either $G$52, the subgroup of squares, or $G$53. Equivalently, among subgroups contained in the square subgroup, only the full square subgroup works (Ma et al., 2020).

For general groups with normal square-free connection sets, the analogue persists in a modified form. A subgroup $G$54 is a perfect code in a Cayley sum graph $G$55 if and only if $G$56 is a left transversal of $G$57 in $G$58, subject to normality constraints on $G$59. This leads to structural restrictions involving the core of $G$60, conjugacy classes in the connection set, and centrality obstructions in connected graphs. The resulting theory classifies the relevant Cayley sum graphs for abelian groups, dihedral groups, and $G$61 (Zhang, 2022).

The symmetric and alternating groups behave very differently in the sum-graph setting. In Cayley sum graphs of $G$62 and $G$63, there is no proper subgroup perfect code: the only subgroup perfect codes are the whole groups themselves (Shaw et al., 5 Sep 2025). This contrasts sharply with the ordinary Cayley-graph theory, where many subgroups of $G$64-related groups are perfect codes.

A related but distinct line studies subgroup sum graphs $G$65 and extended subgroup sum graphs $G$66, where $G$67 is part of the graph definition rather than the code. In that framework, the existence of perfect codes is characterized by involutions in cosets with squares landing in $G$68, and the code-perfect Dedekind groups are exactly $G$69 and $G$70 with $G$71 abelian of odd order (Ma et al., 2024). This is not the same notion as a subgroup perfect code in a Cayley graph, but it shows how subgroup-controlled adjacency produces parallel classification problems.

6. Classification results and current research directions

At the level of whole groups, one important global notion is that of a code-perfect group: a finite group in which every subgroup is a subgroup perfect code. This happens exactly when the group has no element of order $G$72 (Chen et al., 21 Jul 2025). At the opposite extreme, among groups of composite order, those with no nontrivial proper subgroup perfect code are exactly the cyclic $G$73-groups and the generalized quaternion $G$74-groups (Chen et al., 2019).

The $G$75-local structure remains the main organizing principle in recent classification work. For extraspecial $G$76-groups, a subgroup $G$77 is a perfect code if and only if $G$78 is non-abelian, or $G$79 is abelian and not normal, or $G$80 is a maximal abelian subgroup of the plus-type group $G$81; the same paper extends the classification to finite groups whose Sylow $G$82-subgroup is extraspecial (Jingjian et al., 10 Feb 2025). For $G$83-groups with $G$84, and more generally for finite groups with abelian Sylow $G$85-subgroups, subgroup perfect codes are characterized by Frattini-subgroup conditions such as

$G$86

where $G$87 is a Sylow $G$88-subgroup of the subgroup under study and $G$89 is a Sylow $G$90-subgroup of the ambient group (Chen et al., 21 Jul 2025).

Maximal subgroups have recently become a central test case. For $G$91-subgroups, a new local-complement criterion states that $G$92 is a perfect code if and only if every $G$93 with $G$94 yields a complement of order $G$95 in $G$96. This criterion underlies systematic results for point stabilizers in many primitive and quasiprimitive groups, for maximal subgroups of $G$97, and for almost simple groups with socle $G$98; it also leaves open a full characterization for all almost simple groups and for maximal subgroups of $G$99 (Qiao et al., 31 Jul 2025).

The present landscape therefore combines three layers. The first is the classical Cayley-graph theory, where subgroup perfect codes are inverse-closed transversals and tilings. The second is the coset-graph theory of pairs $H$00, where the notion extends to arbitrary vertex-transitive graphs and merges with the theory of $H$01-regular sets. The third is the family of sum-graph and subgroup-sum-graph variants, where squares, nonsquares, and normal subsets replace difference sets. A plausible implication is that the long-term classification problem will continue to reduce to $H$02-local structure, normalizers, and quotient constructions, even when the ambient graph model is no longer a Cayley graph in the strict sense.