Cyclic Subgroup Lattice Overview

• Cyclic subgroup lattice is a poset representing each cyclic subgroup of a finite group, structured by inclusion and divisibility.
• It connects group properties with combinatorial and graph-theoretic invariants, facilitating analysis through power-type graphs and algorithmic reconstruction.
• Algorithmic methods enable the reconstruction of power graphs from the lattice, highlighting its significance in group classification and isomorphism testing.

A cyclic subgroup lattice encodes the inclusion structure of all cyclic subgroups of a finite group and provides a framework for connecting combinatorial, poset-theoretic, and graph-theoretic invariants to group structure. The Hasse diagram of this lattice, frequently termed the "cyclic subgroup graph," captures cover relations and underpins the algebraic-to-combinatorial correspondence emerging in foundational and recent research. The subject has seen renewed focus via its central role in power-type graph theory, algorithmic group isomorphism, and the lattice-theoretic characterization of group properties.

1. Definition and Basic Properties

Let $G$ be a finite group, and denote by $\langle g \rangle$ the cyclic subgroup generated by $g \in G$. The cyclic subgroup lattice is defined as

$$\mathcal{L}_c(G) = \{ \langle g \rangle : g \in G \}$$

with the partial order given by inclusion: $\langle g \rangle \leq \langle h \rangle$ iff $\langle g \rangle \subseteq \langle h \rangle$. Equivalently, $\langle g \rangle \leq \langle h \rangle$ iff $|\langle g \rangle|\,|\,|\langle h \rangle|$. The minimal element is $\{e\}$, and maximal elements correspond to the maximal cyclic subgroups of $G$.

The Hasse diagram of $\mathcal{L}_c(G)$, also termed the cyclic–subgroup graph $\Gamma_{\mathrm{cyc}}(G)$, consists of vertices indexed by cyclic subgroups, with edges representing cover relations: $H_1 \lessdot H_2$ iff $H_1 < H_2$ and there is no cyclic $K$ with $H_1 < K < H_2$ (Sharma et al., 20 Sep 2024).

$\mathcal{L}_c(G)$ is always a meet semilattice since intersections of cyclic subgroups remain cyclic. However, it need not be a lattice (i.e., not every pair of subgroups has a join in $\mathcal{L}_c(G)$), except in the case when $G$ itself is cyclic.

2. Structural Lattice-Theoretic Features

Cyclic subgroup lattices exhibit structural rigidity reflecting underlying group-theoretic properties. For a cyclic group $G \cong \mathbb{Z}_n$, the lattice $\mathcal{L}_c(G)$ is isomorphic to the lattice of positive divisors of $n$ under divisibility, and every subgroup is cyclic. In this context, the lattice is distributive and complemented; in particular, it is isomorphic to the product of chains $[0,a_i]$ for $n = \prod_{i=1}^k p_i^{a_i}$.

In contrast, for noncyclic groups, $\mathcal{L}_c(G)$ generally fails to be distributive and complemented. Intervals in the lattice correspond to divisibility intervals in the factorization of $|G|$ for cyclic $G$, and to more complex combinatorial posets for non-cyclic $G$ (Sharma et al., 20 Sep 2024).

Ore's theorem asserts that $G$ is cyclic if and only if the entire subgroup lattice $\mathcal{L}(G)$ is distributive (Palcoux, 2017). For $\mathcal{L}_c(G)$, distributivity characterizes the case where the lattice is the full divisor lattice of $|G|$, equating to $G$ being cyclic (Sharma et al., 20 Sep 2024).

3. Correspondence With Power-Type Graphs

A profound development is the explicit two-way correspondence between cyclic subgroup lattices and power-type graphs—specifically, the enhanced power graph $\mathrm{EPow}(G)$, the power graph, the directed power graph, and the difference graph. The main result establishes:

• The (unlabeled) enhanced power graph $\mathrm{EPow}(G)$ uniquely determines $\mathcal{L}_c(G)$.
• Conversely, the labeled lattice $\mathcal{L}_c(G)$ uniquely determines $\mathrm{EPow}(G)$.

This correspondence operates purely at the level of cliques and clique-intersections in the graph and chains and cover relations in the lattice, independent of the explicit group operation. The process involves constructing the local divisor posets for each maximal cyclic subgroup and then forming the quotient poset by identifying subgroups according to clique-intersection data. Algorithmic procedures for reconstructing each object from the other have been established, with polynomial-time complexity for the relevant enumeration and identification steps (Mirzargar et al., 15 Nov 2025).

Other power-type graphs—such as the ordinary power graph or directed power graph—can similarly be obtained from $\mathcal{L}_c(G)$ through minor alterations in the gluing of cliques and orientation of edges. As a result, all power-type graphs encode exactly the same information as $\mathcal{L}_c(G)$, yielding a combinatorial equivalence of invariants (Mirzargar et al., 15 Nov 2025).

4. Cyclic Subgroup Graph: Combinatorial and Graph-Theoretic Invariants

The cyclic–subgroup graph $\Gamma_{\mathrm{cyc}}(G)$ is the undirected Hasse diagram of $\mathcal{L}_c(G)$ (Sharma et al., 20 Sep 2024). Key properties include:

• Bipartiteness and connectivity: $\Gamma_{\mathrm{cyc}}(G)$ is always bipartite, connected, and perfect.
• Diameter and structure: For $G \cong \mathbb{Z}_n$, with $n = \prod_{i=1}^k p_i^{a_i}$, the diameter is $\sum_{i=1}^k a_i$.
• Regularity: The graph is regular iff $G$ is cyclic and $n$ is square-free, in which case it is the $k$-cube (Boolean hypercube).
• Eulerian property: The graph is Eulerian iff $G$ is cyclic of square-free, even rank.

The number of vertices $v(G) = |\mathcal{L}_c(G)|$ and the number of edges $e(G)$ can be computed in closed form for several families of groups. For $G = \mathbb{Z}_n$,

$e(G) = \sum_{i=1}^k a_i \prod_{j \neq i} (a_j + 1)$

and for $G = D_{2n}$ (dihedral) or $Dic_n$ (dicyclic):

$e(G) = \sum_{i=1}^k a_i \prod_{j\neq i}(a_j+1) + n$

For $Q_{2^r}$ (generalized quaternion),

$e(Q_{2^r}) = 2^{r-2} + r-1$

(Sharma et al., 20 Sep 2024).

5. Lattice-Theoretic Characterizations and Classical Results

Distributivity and complementation in $\mathcal{L}_c(G)$ correspond to strong structural properties. $\mathcal{L}_c(G)$ is distributive if and only if $G$ is cyclic. It is Boolean (complemented and distributive) if and only if $G \cong \mathbb{Z}_n$ with $n$ square-free, yielding a hypercube lattice (Sharma et al., 20 Sep 2024).

Ore's theorem provides the group-theoretic counterpart: $G$ is cyclic if and only if its full subgroup lattice $\mathcal{L}(G)$ is distributive. Generalizations to distributivity on intervals yield wider consequences: a distributive interval $[H,G]$ ensures that there exists $g\in G$ with $⟨H,g⟩ = G$, and in the dual setting, that $[H,G]$ is "linearly primitive" in the sense of representation theory (Palcoux, 2017). Notably, $\mathcal{L}_c(G)$ for noncyclic groups fails distributivity, as exemplified by $V_4 \cong \mathbb{Z}_2 \times \mathbb{Z}_2$.

6. Algorithmic, Classification, and Open Directions

Reconstruction algorithms for moving between $\mathcal{L}_c(G)$ and the enhanced power graph have polynomial-time complexity and can be implemented within computational group theory packages (Mirzargar et al., 15 Nov 2025). This bidirectional equivalence reduces several classification problems for power-type graphs—such as determining if groups are isomorphic given their power graphs—to the corresponding problems for their cyclic subgroup lattices.

If a natural augmentation of the power graph is defined (e.g., by remembering vertex-labels of element orders), it is plausible that the full subgroup lattice $\mathcal{L}(G)$ could be reconstructed, extending from cyclic subgroups to all subgroups (Mirzargar et al., 15 Nov 2025).

A central open problem concerns the extent to which cyclic subgroup lattices and their associated graphs characterize group structure up to isomorphism ("which groups are determined by their power-type graph?"). This reduces to lattice-isomorphism classification, with consequences for group invariants, automorphism groups, and isomorphism testing. Known results for abelian, $p$-groups, and certain families like dihedral or quaternionic groups transfer equivalently to the cyclic subgroup graph framework (Mirzargar et al., 15 Nov 2025).

7. Illustrative Examples

Cyclic Groups $\mathbb{Z}_n$:

• The cyclic subgroup lattice is the full divisor lattice of $n$.
• For $n = p^k$, the lattice is a chain and $\Gamma_{\mathrm{cyc}}(\mathbb{Z}_{p^k})$ is a path.
• For $n = pq$ ($p \neq q$), the lattice forms a square and the cyclic subgroup graph is $C_4$ (4-cycle).

Symmetric Group $S_3$:

• Cyclic subgroups are: $\{e\}$ (order 1), three subgroups of order 2, one of order 3.
• The Hasse diagram includes a 3-clique on elements of order 3, and three 2-cliques for each order 2 subgroup glued at $\{e\}$.
• The enhanced power graph and the cyclic subgroup lattice can be recovered from one another via the clique structure and intersections (Mirzargar et al., 15 Nov 2025).

Non-Cyclic Example $V_4$:

The lattice fails distributivity, matching the group-theoretic non-cyclicity; the order-2 subgroups form a three-branching structure above $\{e\}$, with no mutually covering relations (Palcoux, 2017).

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The combinatorial equivalence between the cyclic subgroup lattice and power-type graphs provides a powerful framework for studying and classifying finite groups, enabling translation between group-theoretic, poset-theoretic, and graph-theoretic perspectives. The cyclic subgroup lattice encodes not only direct subgroup structure, but through its interrelation with graph invariants, also connects to broader algorithmic and structural group theory.