Conjugacy Classes of Cyclic Subgroups

• Conjugacy classes of cyclic subgroups are defined by invariant multisets of elementary divisors and rational canonical forms in various group contexts.
• The topic examines classifications in finite groups, matrix groups like GL(n, Fq), and p-groups using eigenstructure and fusion system methodologies.
• It provides actionable insights for applications in representation theory, combinatorics, coding theory, and algebraic topology.

The conjugacy classes of cyclic subgroups in group-theoretic settings are a central invariant with implications across finite group theory, representation theory, algebraic combinatorics, and applications such as coding theory. Classification results for conjugacy classes of cyclic subgroups are especially fine in matrix groups over finite fields, $p$-groups, and various geometric and combinatorial group constructions.

1. Conjugacy in Finite Groups and Matrix Groups

Let $G$ be a finite group. Two subgroups $H,K<G$ are conjugate if there exists $g\in G$ such that $K=g^{-1}Hg$. Within $G$, cyclic subgroups play a special role due to their simplicity and universality. Classifying cyclic subgroups up to conjugacy requires understanding the possible generators and the way their group-theoretic invariants (such as minimal and characteristic polynomials in linear groups) control the orbit structure under conjugation.

In the general linear group $\mathrm{GL}_n(\mathbb{F}_q)$, the complete classification of conjugacy classes of cyclic subgroups is rooted in rational canonical forms and the associated multiset of elementary divisors for generators. Let $S_A=\langle A\rangle$ and $S_B=\langle B\rangle$. Then $S_A$ and $G$0 are conjugate in $G$1 if and only if $G$2 and $G$3 have matching multisets of elementary divisors, with matched degrees, exponents, and crucially, the same orders of the irreducible polynomials dividing $G$4, where $G$5 (Manganiello et al., 2011):

• $G$6 is conjugate to $G$7 if and only if:
  • $H,K<G$2,
  • $H,K<G$3,
  • $H,K<G$4.

Each conjugacy class is uniquely specified by the multiset $H,K<G$5, where $H,K<G$6 runs over monic irreducible divisors of $H,K<G$7 and $H,K<G$8 denotes their multiplicity. For a generator $H,K<G$9, the rational canonical form is block-diagonal with companion matrices $g\in G$0, and each cyclic subgroup of order $g\in G$1 is determined (up to conjugation) by such a decomposition.

A purely structural summary:

Invariant	Data Required	Classification Principle
Elementary divisors	Multiset $g\in G$2 with $g\in G$3	All $g\in G$4, degrees, orders coincide
Rational canonical form	Block-diagonal, $g\in G$5 blocks	Uniqueness up to block permutation
Orbit codes (applications)	Conjugacy class determines code type	Reduce to “canonical” cyclic subgroups

No explicit closed form is provided for the number of classes in general dimension; enumeration is by combinatorial enumeration of possible multisets $g\in G$6 as above (Manganiello et al., 2011).

2. Specific Classifications: $g\in G$7 and PGL

For $g\in G$8, the classification of conjugacy classes of completely reducible cyclic subgroups of order $g\in G$9 with $K=g^{-1}Hg$0 proceeds by analyzing the eigenstructure:

• Split (non-central): Diagonalizable over $K=g^{-1}Hg$1 with distinct eigenvalues.
• Non-split (irreducible): Minimal polynomial irreducible of degree 2 ($K=g^{-1}Hg$2 divides $K=g^{-1}Hg$3, not $K=g^{-1}Hg$4, i.e., $K=g^{-1}Hg$5).
• Central (scalar): $K=g^{-1}Hg$6 with $K=g^{-1}Hg$7 an $K=g^{-1}Hg$8th root of unity in $K=g^{-1}Hg$9.

Explicit counts are given (Kumar et al., 2024):

• Non-central split classes:

$G$0

where $G$1 and $G$2.

• Irreducible (non-split) classes:

$G$3

• Central class: $G$4 if $G$5, else $G$6.

For $G$7, conjugacy classes of admissible cyclic subgroups of prime order $G$8 are parameterized by diagonalizable elements up to permutation and scalar shifts, with a piecewise explicit enumeration depending on $G$9 possible cycle lengths in the symmetric group $\mathrm{GL}_n(\mathbb{F}_q)$0. There is also a duality (“association”) linking the counts in $\mathrm{GL}_n(\mathbb{F}_q)$1 and in complementary dimensions and a link with Gale duality for point-sets (Marinatto, 2020).

3. Fusion Systems and Saturated Conjugacy

For saturated fusion systems $\mathrm{GL}_n(\mathbb{F}_q)$2 on a finite $\mathrm{GL}_n(\mathbb{F}_q)$3-group $\mathrm{GL}_n(\mathbb{F}_q)$4, Park (Park, 2013) shows that the number of $\mathrm{GL}_n(\mathbb{F}_q)$5-conjugacy classes of cyclic subgroups in $\mathrm{GL}_n(\mathbb{F}_q)$6 equals the rank of a square matrix $\mathrm{GL}_n(\mathbb{F}_q)$7 whose entries count orbits or double cosets corresponding to $\mathrm{GL}_n(\mathbb{F}_q)$8. Concretely, for compatible group models $\mathrm{GL}_n(\mathbb{F}_q)$9 with $S_A=\langle A\rangle$0, one has:

$S_A=\langle A\rangle$1

with $S_A=\langle A\rangle$2, over representatives $S_A=\langle A\rangle$3 for $S_A=\langle A\rangle$4-conjugacy classes.

The same holds using more abstract constructions such as characteristic $S_A=\langle A\rangle$5-bisets or the characteristic idempotent in the double Burnside ring, providing an operator-theoretic and Burnside ring framework for counting conjugacy classes in the fusion-theoretic setting.

4. Cyclic Subgroups and Coverings in $S_A=\langle A\rangle$6-Groups

In the context of finite $S_A=\langle A\rangle$7-groups $S_A=\langle A\rangle$8 of order $S_A=\langle A\rangle$9 and nilpotence class $S_B=\langle B\rangle$0, the function $S_B=\langle B\rangle$1, counting conjugacy classes of maximal cyclic subgroups, has strict lower bounds depending linearly on $S_B=\langle B\rangle$2:

$S_B=\langle B\rangle$3

Equality holds in specific abelian (class 1) and certain extensions, and generally this reveals the structural impact of nilpotence and group size on the diversity of cyclic subgroup classes (Bianchi et al., 2022). For metacyclic $S_B=\langle B\rangle$4-groups, explicit formulas for $S_B=\langle B\rangle$5 are derived, and except for a few exceptional negative-type cases (dihedral, quaternion, semidihedral, and two subfamilies), always satisfy $S_B=\langle B\rangle$6, with strict inequalities in non-exceptional cases (Bianchi et al., 2022).

Under group extension and via direct/semi-direct products, $S_B=\langle B\rangle$7 obeys monotonic (or in select cases strict multiplicative) behavior, with full formulae for Frobenius and abelian cases (Bianchi et al., 2022, Bianchi et al., 2022).

5. Counting in $S_B=\langle B\rangle$8-Compact Groups and Homotopical Context

In the setting of $S_B=\langle B\rangle$9-compact groups, the cardinality of the set of conjugacy classes of (homotopy classes of) maps from $S_A$0 to $S_A$1—which correspond bijectively to conjugacy classes of cyclic subgroups of order dividing $S_A$2 in the sense of group theory—can be recast as the Weyl group orbit count of the reflection representation lattice mod $S_A$3 (Cantarero et al., 13 Oct 2025):

$S_A$4

with $S_A$5 the character lattice and $S_A$6 the Weyl/reflection group. Via Burnside’s lemma and the character theory of $S_A$7, this admits closed formulas (notably Solomon’s product-formula in the non-modular case).

6. Illustrative Examples

• In $S_A$8, the irreducible divisors of $S_A$9 identify two conjugacy classes of cyclic subgroups of order 7: those generated by $G$00 and $G$01 (Manganiello et al., 2011).
• In $G$02, cyclic subgroups of order 3 correspond either to three companion blocks of $G$03, or two of $G$04, yielding distinct conjugacy classes.

7. Applications and Significance

The classification of conjugacy classes of cyclic subgroups provides deep insight into group structure, impacts the computation of class functions and representation modules, and is directly applicable to orbit codes in network coding theory (where conjugacy-class representatives parameterize code equivalence), as well as to the study of group cohomology and finite group actions in homotopy theory (Manganiello et al., 2011, Cantarero et al., 13 Oct 2025). For $G$05-groups and their fusion systems, structural results underpin questions of normal coverings, extension theory, and the asymptotic analysis of group invariants (Bianchi et al., 2022, Park, 2013).

The precise classification results enable algorithmic computation of subgroup invariants and inform constructive applications in geometric group theory, combinatorics, and algebraic coding.