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Finite permutation groups with quasi-semiregular elements

T0 review · 2 major / 4 minor · reviewed 2026-08-06 · deepseek-v4-flash

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Pith's one-line read Finite primitive groups containing an element that fixes exactly one point belong to only five O'Nan-Scott types, and for alternating and sporadic socles the possibilities are completely listed.

desk verdict Solid O'Nan-Scott reduction and alternating classification, but the sporadic proof has an unstated completeness gap and Table 2 has wrong indices. read the letter →

arxiv 2507.12866 v1 pith:6AX3IJDK submitted 2025-07-17 math.GR

Michael Giudici , Luke Morgan , Cheryl E. Praeger This is my paper · ORCID
classification math.GR MSC 20B0520B15
keywords quasi-semiregularelementsprimitivepermutationgroupsO'Nan-Scotttheoremalmostsimplealternatingsporadicderangementssemiregularpermutations
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

A quasi-semiregular element of a permutation group fixes exactly one point and moves all others in cycles of equal length. The paper sets out to determine which finite primitive permutation groups contain such an element. It proves that only the O'Nan-Scott types HA, AS, PA, SD, and CD can occur, and reduces the PA and CD cases to existence in the base group. For maximal diagonal-type groups the existence is governed by a clean numerical criterion: the diagonal degree $k$ must be prime and coprime to $|T|$. It then gives complete classifications for almost simple groups whose socle is alternating or sporadic, listing the maximal point stabilisers $H$ and prime orders $p$ for which the coset action on $[G:H]$ admits a quasi-semiregular element.

What carries the argument

The load-bearing object is the quasi-semiregular element itself: a prime-order element whose cycle type is $1^1p^{(n-1)/p}$ on $n$ points. The arguments run on four mechanisms. Lemma 2.1 says quasi-semiregularity survives passage to any system of imprimitivity, so impossible types can be ruled out and product actions reduced. Lemma 2.3 and Corollary 2.5 convert the condition into computable normaliser and centraliser statements: for $x\in G_\alpha$ of prime order, $x$ is quasi-semiregular if and only if $x^G\cap G_\alpha = x^{G_\alpha}$ and $C_G(x)=C_{G_\alpha}(x)$, while Corollary 2.7 gives the subgroup version $N_G(\langle x\rangle)=N_{G_\alpha}(\langle x\rangle)$ and $K^G\cap G_\alpha = K^{G_\alpha}$. Theorem 5.5 solves the diagonal case by showing that a quasi-semiregular element of $T^k.(\mathrm{Out}(T)\times\mathrm{Sym}(k))$ must come from a $k$-cycle in the wreath factor, with Thompson's theorem controlling fixed points of automorphisms. The sporadic tables are generated by reading the same character-theoretic criteria off stored fusion data.

What would settle it

Recalculate the sporadic tables with an independent implementation of the fusion data: any maximal subgroup row not appearing in Tables 6-9 whose coset action contains an element with exactly one fixed point would falsify Theorem 1.5. Equally, any primitive group of degree below 4096 of O'Nan-Scott type HS, HC, or TW containing a quasi-semiregular element would falsify Theorem 1.1, and any maximal SD group with $k$ prime and $\gcd(|T|,k)=1$ lacking one would falsify the 'if' direction of Theorem 1.1(a).

Watch

Extended reading notes

Core claim

The central discovery is that quasi-semiregular elements force a primitive group into a very restricted structural class, and for large families the existence question becomes a checkable condition in a smaller group. Theorem 1.1 states that a finite primitive permutation group containing a quasi-semiregular element must have O'Nan-Scott type HA, AS, PA, SD, or CD; for maximal SD groups with socle $T^k$, such elements exist if and only if $k$ is prime and $\gcd(|T|,k)=1$, in which case every quasi-semiregular element is conjugate to an element of $\mathrm{Sym}(k)$ of order $k$; and for maximal product actions of type PA or CD, existence is equivalent to existence in the base group. Theorem 1.2 shows that every $\frac{3}{2}$-transitive affine group contains one. For almost simple groups with alternating socle, Theorem 1.4 lists the maximal subgroups (set stabilisers, partition stabilisers, affine and projective-line subgroups, and a small table of exceptional embeddings) together with the prime orders that work; Theorem 1.5 does the same for the twenty-six sporadic simple groups through explicit tables. Corollary 1.6 records that having two distinct conjugacy classes of quasi-semiregular subgroups of the same prime order is a rare event, with a short list of exceptions.

Load-bearing premise

The classification rests on the completeness and correctness of the stored computer data: the character tables with their fusion maps, the complete list of maximal subgroups of the Monster, and the primitive-group database used for small exceptional affine cases.

Editorial extensions

If this is right

  • Any primitive group of type HS, HC, or TW has no quasi-semiregular elements, so the search in primitive groups reduces to affine and almost simple actions.
  • For maximal product-action groups, existence in $H\wr\mathrm{Sym}(\ell)$ is equivalent to existence in $H$ on $\Delta$, so the PA and CD families are settled once the base group is known.
  • For maximal SD groups, quasi-semiregular elements exist exactly when $k$ is prime and $\gcd(|T|,k)=1$, and they are all conjugates of the $k$-cycle in $\mathrm{Sym}(k)$; non-maximal SD groups can fail even under that condition.
  • Every $\frac{3}{2}$-transitive affine group contains a quasi-semiregular element, so the remaining affine question is exactly Problem 1.3: which primitive affine groups have none.
  • All alternating-socle and sporadic-socle almost simple primitive actions with a quasi-semiregular element are listed in tables, and Corollary 1.6 lists the only cases with two conjugacy classes of quasi-semiregular subgroups of the same prime order.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • Because quasi-semiregularity descends to every system of imprimitivity, the primitive classification is likely to be the first layer of a full classification for transitive groups: one could pull the element up through block systems and check which overgroups of the stabiliser preserve it.
  • The tables should give graph theorists explicit new quasi-$m$-Cayley graphs: any row with a quasi-semiregular $p$-element and stabiliser $H$ yields a candidate vertex-transitive graph whenever the coset action can be realised as a graphical regular cover.
  • The subnormaliser criterion in Theorem 3.3 suggests that quasi-semiregular elements are the orbit-level shadow of picky elements, so the classification may feed the local-global character-theoretic questions that motivated subnormalisers.
  • Combining the alternating and sporadic classification with the announced Lie-type work would give a complete list of almost simple primitive groups with quasi-semiregular elements, leaving only the affiliate primitive groups of Problem 1.3.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

2 major / 4 minor

Summary. The paper studies finite permutation groups containing a quasi-semiregular element, i.e., an element with a unique fixed point and all its other cycles of equal length. The main results are: (1) a reduction theorem (Theorem 1.1) showing that a finite primitive group containing such an element must have O'Nan-Scott type HA, AS, PA, SD, or CD, with precise existence criteria for maximal SD groups and for maximal product actions; (2) a proof that every 3/2-transitive affine group contains a quasi-semiregular element (Theorem 1.2); (3) a complete classification of almost simple primitive groups with alternating socle that admit quasi-semiregular elements (Theorem 1.4, Tables 1 and 2); and (4) a classification for almost simple groups with sporadic socle (Theorem 1.5, Tables 6-9), obtained largely by computation with GAP's character table library, with special arguments for the Baby Monster and the Monster. The paper also derives a corollary describing when a primitive group with alternating or sporadic socle has at least two conjugacy classes of quasi-semiregular subgroups of the same prime order (Corollary 1.6). The structural sections (Sections 2, 5, and 6) are coherent and the main non-computational arguments appear sound.

Significance. If the computational parts are reliable, this is a valuable contribution to permutation group theory. The reduction to O'Nan-Scott types and the explicit classification for alternating and sporadic socles are significant advances on a natural analogue of the Polycirculant Conjecture, with connections to quasi-Cayley graphs and strongly p-embedded subgroups. The paper contains several clean, checkable theoretical results, notably Theorem 5.5 on diagonal groups and Theorem 5.9 on twisted wreath groups, and it explicitly identifies the remaining open problems for affine groups and groups of Lie type. The use of standard GAP character-table data and the recent classification of maximal subgroups of the Monster is appropriate, but the lack of reproducible scripts and the terseness of some manual computational arguments mean the 'if and only if' claims for sporadic socles are not yet fully verifiable from the manuscript alone.

major comments (2)
  1. [Section 7.2 (Lemma 7.2)] The proof of Lemma 7.2 does not currently establish exhaustiveness for the Monster. After the reductions, the text states that each maximal subgroup with character table not stored in [6] will be considered 'in turn', but only two such subgroups are treated: H = 2^{5+10+20}.(S3 × L5(2)) and H = 59:29. The manuscript does not state that these are the only maximal subgroups of the Monster whose character tables are absent from CTblLib, nor does it enumerate the maximal subgroups from [14,13] that fall into this category. Since Theorem 1.5 is an 'if and only if' statement over all maximal subgroups of the Monster, this missing enumeration is load-bearing. Please either list all such subgroups and give the argument for each, or provide a machine-checkable computation using the explicit constructions of [14,13] that covers all maximal subgroups of the Monster.
  2. [Sections 7.1 and 7.2 (Lemmas 7.1 and 7.2)] The correctness of Theorem 1.5 depends on the completeness and correctness of the maximal subgroup lists and fusion maps in the GAP Character Table Library [6] for all sporadic groups except the Monster, and on the completeness of the maximal subgroup classification of the Monster from [14,13]. The manuscript uses functions like HasMaxes and NamesOfFusionSources to assert this completeness, but it does not cite a source for the completeness of the CTblLib data for the relevant groups, and it does not provide the verification scripts. In addition, the manual arguments for the exceptional fusion-map cases (the Baby Monster with H = (2^2 × F4(2)).2 in Lemma 7.1, and the two Monster cases in Lemma 7.2) verify the conditions of Corollary 2.5 only partially as written. Please state explicitly that the external data are complete (with references), and make the GAP/Magma scripts used for Lemmas 7.1-7.2 and Theorem 4.1 available, so that the computational claims are reproducible and the 'if and only if' assertions can be independently checked.
minor comments (4)
  1. [Table 2] The |G:H| column of Table 2 contains several numerical errors: for example, |Alt(7):PSL(3,2)| = 15, not 120; |Alt(8):AGL(3,2)| = 15, not 120; |Alt(9):PΓL(2,8)| = 120, not 280; and |Alt(11):M11| = |Alt(12):M12| = 2520, not 362880. These indices are not used in the proof of Theorem 1.4, but the table should be corrected and independently verified before publication.
  2. [Section 3.2] In the discussion after Theorem 3.3, the example x = (1,2,3,4)(5,6) is said to be picky in G = Alt(7), but the earlier statement defines picky elements as lying in a unique Sylow p-subgroup; the example has order 4, so the phrase 'picky p-element' should be clarified to avoid confusion with the prime order assumption used elsewhere.
  3. [Section 6, Proposition 6.4] The notation '1 p p m−1' for cycle types is ambiguous in the printed text; it should be written as 1^p p^{m-1} (and similarly p^m) to match the standard convention for cycle types.
  4. [Section 4.1] In the proof of Theorem 4.1 for the G2(q) case, the expression 'q3−1 q−1' is a typographical rendering of (q^3-1)/(q-1); this should be typeset as a fraction for readability.

Circularity Check

0 steps flagged · score 2.0 of 10

No significant circularity: the derivation chain is genuine, with only computational-completeness and table-accuracy caveats.

full rationale

The paper's central derivations are not circular. The fixed-point formula (Lemma 2.4), the quasi-semiregularity criterion (Corollary 2.5), the imprimitivity reduction (Lemma 2.1), and the product-action/diagonal-type analyses of Section 5 prove the O'Nan-Scott reduction rather than assuming it. The alternating-group classification of Theorem 1.4 is derived from Jordan's theorem, the Sylow arguments of Lemma 2.3, and Jones's classification of primitive groups containing a cycle, with each case then verified by normalizer computations. The sporadic classification of Theorem 1.5 is a computational enumeration using GAP's CTblLib and the recently completed Monster maximal-subgroup lists; this creates a completeness risk, not a circularity, because no parameter is fitted and no conclusion is fed back as an input. The paper cites the authors' own prior work, including the 3/2-transitive classification of Liebeck, Praeger, and Saxl and the O'Nan-Scott division of Praeger, but these are independent published theorems whose assumptions do not include the target result, so the citations are real evidence rather than load-bearing self-reference. Two caveats should be recorded but do not affect the circularity score. First, in the proof of Lemma 7.2 the paper says 'We will consider each maximal subgroup H in turn' but then explicitly treats only H = 2^{5+10+20}.(S3 x L5(2)) and H = 59:29, without enumerating all non-GAP maximal subgroups from [14,13]; this is an incomplete demonstration of exhaustiveness, though not a circular step. Second, Table 2 contains apparent index errors, for example row 4 lists |Alt(11):M11| as 362880, whereas the index is 2520; such errors affect table reliability but not the logical derivation. Overall, the classification is derived, not assumed, and the paper scores near the non-circular end of the scale, with minor self-citation that is not load-bearing.

Assumptions & free parameters 0 free parameters · 9 assumptions · 0 invented entities

The central claims rest on standard classification theorems of finite group theory plus completeness assumptions for computational data. No numerical free parameters are fitted, and no new entities are introduced. The main external dependence is the completeness of character table fusion data and maximal subgroup lists in GAP and in the recent Monster work [13,14]; the paper explicitly flags only one missing fusion map in the Baby Monster case and handles it manually.

assumptions (9)
  • domain assumption Classification of the Finite Simple Groups (CFSG)
    Invoked through O'Nan-Scott theory, the 3/2-transitive classification [36], and the strongly p-embedded subgroup classification [22]. The sporadic and alternating classifications inherit CFSG.
  • domain assumption O'Nan-Scott theorem, specifically the eight-type division of finite primitive permutation groups
    Foundational for Theorem 1.1 and for the reductions in Section 5; assumed as background from [47].
  • domain assumption Liebeck-Praeger-Saxl classification of 3/2-transitive groups [36, Corollary 2]
    Quoted as Theorem 4.2. Theorem 1.2 is proved by checking each case in this classification.
  • domain assumption Jones classification of primitive permutation groups containing a p-cycle [29, Theorem 1.2]
    Used in the proof of Theorem 6.5 to enumerate the possible primitive maximal subgroups containing a p-cycle.
  • standard math Thompson fixed-point-free automorphism theorem [54]
    Used in Theorems 5.5, 5.9, and 5.10 to produce nonidentity fixed points or non-fixed points for automorphisms of prime order.
  • standard math Manning fixed-point formula, as stated in Theorem 2.6 and Corollary 2.7
    Grounds the criteria in Corollaries 2.5 and 2.7 that are used throughout the sporadic computations and in some alternating-group arguments.
  • domain assumption Completeness and correctness of GAP CTblLib data [6] and maximal subgroup classifications for sporadic groups, including the Monster from [14,13]
    Lemmas 7.1 and 7.2 decide the sporadic cases by checking stored character tables, maximal subgroup lists, and fusion maps. Missing or incorrect fusion data would change the classification.
  • domain assumption Correctness of the Magma primitive group database [10] and of Magma computations for exceptional groups
    Used in Theorem 4.1, Theorem 1.2 case (5), and Proposition 6.1 to confirm small exceptional cases. No scripts or logs are provided.
  • domain assumption Liebeck-Praeger-Saxl classification of maximal subgroups of alternating and symmetric groups [35]
    Used in Theorem 6.5 to determine when listed subgroups such as AGL(1,p) or PGammaL(2,8) are maximal in Alt(n) or Sym(n).

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Pith. "Pith review of Finite permutation groups with quasi-semiregular elements." pith.science (2026). https://pith.science/paper/6AX3IJDK

@misc{pith2026250712866,
  author       = {Pith},
  title        = {Pith review of: Finite permutation groups with quasi-semiregular elements},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/6AX3IJDK}},
  note         = {Machine review of arXiv:2507.12866}
}
abstract

A quasi-semiregular element in a permutation group is an element that has a unique fixed point and acts semiregularly on the remaining points. Such elements were first studied in the context of automorphisms of graphs and occur naturally in many families of permutation groups, such as Frobenius and Zassenhaus groups. They also arise in the context of groups with a strongly $p$-embedded subgroup. We investigate the question of which finite permutation groups contain quasi-semiregular elements, with particular attention to the primitive permutation groups. We determine the O'Nan-Scott types of primitive groups that can contain quasi-semiregular elements and reduce the question to the affine and almost simple cases. In the almost simple case, we obtain a complete classification when the socle is alternating or sporadic.

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