bibkey: lazorec2025subgroupconjugacy
authors: Mihai-Silviu Lazorec and Marius Tărnăuceanu
year: 2025
title: On the number of conjugacy classes of subgroups of a finite group
doi: 10.48550/arXiv.2508.14267
url: https://arxiv.org/html/2508.14267v2
claim: Proposition 2.2(ii) and its proof give the whole subgroup-lattice count 2n+p−1 and inner-conjugacy class count 2n for the cyclic split extensions G_{p,q,n}; Section 4 Problem 4.1 asks whether every rational a/b with integers 1≤a<b is attained by an actual finite group.
strata_touched:
  - D5/S3/FiniteGroups/CyclicPrimePowerAction
  - D5/S3/FiniteGroups/RationalSubgroupConjugacyAttainment
  - D5/S3/FiniteGroups/SubgroupConjugacyRatio
license: CC-BY-4.0
triage: anchor

Subgroup conjugacy ratios of finite groups

The primary source is arXiv:2508.14267v2, posted 9 August 2026; the first version was posted 19 August 2025. The title, authors and CC BY 4.0 license are given on the arXiv abstract page and HTML. The versioned HTML is the source for the locators below.

Verified locator

• DOI: 10.48550/arXiv.2508.14267.
• Primary URL: https://arxiv.org/html/2508.14267v2.
• Version and license: https://arxiv.org/abs/2508.14267v2.
• Section 1: https://arxiv.org/html/2508.14267v2#S1. All groups in the paper are finite. Here $L(G)$ is the entire subgroup lattice, $k^{\prime }(G)$ is its number of conjugacy classes, and $d^{\prime }(G)=k^{\prime }(G)/|L(G)|$. The section states multiplicativity of $d^{\prime }$ for direct products of groups of pairwise coprime orders.
• Section 2, Lemma 2.1(ii) and Proposition 2.2(ii): https://arxiv.org/html/2508.14267v2#S2. Lemma 2.1(ii) describes a nonabelian split extension $G_{p,q,n}=N⋊P$, where $N\cong C_p$, $P\cong C_{q^{n-1}}$, $[N,\Phi (P)]=1$, $p,q$ are primes, $q\mid p-1$, and $n\ge 2$. Proposition 2.2(ii) states $d^{\prime }(G_{p,q,n})=2n/(2n+p-1)$.
• The proof of Proposition 2.2(ii), equation S2.Ex12: https://arxiv.org/html/2508.14267v2#S2.Ex12. It explicitly states both counts $k^{\prime }(G_{p,q,n})=2n$ and $|L(G_{p,q,n})|=2n+p-1$.
• Section 2, Corollary 2.4 already gives attainment of $a/(a+1)$; Theorem A gives density of the nilpotent-group values in $[0,1]$.
• Section 4, Problem 4.1: https://arxiv.org/html/2508.14267v2#S4. For integers $1\le a<b$, the authors ask whether some group has $d^{\prime }(G)=a/b$. They propose choosing $b-a$ groups of coprime orders with ratios $(a+i-1)/(a+i)$ and taking their direct product. Problems 4.2 and 4.3 concern different structural and extremal questions.

Scope and parameter correspondence

CyclicPrimePowerAction.split_counts concerns primes p,q, a natural s>0, coprimality of p,q, and an actual ActionData p q s. The latter contains a homomorphism $C_{q^s}⟶\mathrm{Aut}(C_p)$ whose generator image has exact order $q$. SplitGroup A is its actual semidirect product. Set $n=s+1$, $N=C_p$, and $P=C_{q^s}$. Then $n\ge 2$ and the generator acts nontrivially, while its $q$th power acts trivially. The Frattini subgroup of this cyclic prime-power $P$ is generated by that $q$th power, so the action has the source condition $[N,\Phi (P)]=1$. For the source’s primes with $q\mid p-1$, actionData constructs such an actual action using Cauchy’s theorem and the cyclic-generator extension. The primes are distinct, so the count theorem’s coprimality condition holds.

The Lean count theorem works for every supplied actual action in this scope, without assuming the source’s count formulas or its classification. Its whole-lattice bijection gives $2s+p+1=2n+p-1$ subgroups. Its inner-conjugacy quotient bijection gives $2s+2=2n$ classes. Both counts include bottom and top; s=1 is the source boundary case n=2. These numerical statements are literature-attested, while their Lean proofs supply kernel evidence. No mathematical novelty is claimed for them.

SubgroupConjugacyRatio uses precisely the source observable: latticeCount G = Nat.card (Subgroup G) and classCount G = Nat.card (Quotient (MulAction.orbitRel (ConjAct G) (Subgroup G))). This is subgroup conjugacy by elements of the actual group, rather than subgroup isomorphism or conjugacy of elements. coprime_product_counts additionally constructs bijections for both counts separately; its ratio consequence is the known Section 1 identity.

RationalSubgroupConjugacyAttainment.rational_attainment is the repository proof of the universal question, not a theorem asserted by the source. It supplies fresh consecutive-ratio factors of orders coprime to the accumulated group and telescopes them. Its final target has every integer pair $1\le a<b$ and a genuine finite-group witness. Both counts are positive, so $b{k}^{\prime }(G)=a|L(G)|$ is equivalent to $d^{\prime }(G)=a/b$. It assumes no nilpotence, supersolvability, bounded range or reduced fraction.

Literature and open-status qualification

The primary v2 still explicitly poses Problem 4.1. The bounded search qualification in the preregistration reports reading the primary version history and arXiv:2608.08845v1, 2607.05700v1 and 2509.17004v1; screening 24 Lazorec author results and the newest 50 of 101 Tărnăuceanu author results; and title and subgroup-conjugacy searches. No resolution was located within that reported scope. The preregistration also reports repository and issue searches with no duplicate exact target located. Those search outcomes are attributed evidence, not an independently exhaustive absence claim. OpenAlex and Semantic Scholar citation APIs were rate limited; Google and Bing results were unusable. A later or unindexed independent resolution cannot be excluded. No worldwide novelty or priority claim follows from this qualification.

The formal conclusion and its applicability are documented in the Problem 4.1 dossier.