Finite simple groups have many classes of $p$-elements

Cheryl E. Praeger; Luke Morgan; Michael Giudici

Finite simple groups have many classes of $p$-elements

T0 review · 0 major / 7 minor · reviewed 2026-08-12 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗ Review JSON

Pith's one-line read The paper proves that the order of any finite nonabelian simple group is bounded above by an increasing function of the largest number, over all primes $p$, of $\mathrm{Aut}(T)$-classes of elements of $p$-power order.

desk verdict New bound on |T| in terms of Aut-classes of p-elements; proof is detailed and likely correct, with the usual heavy reliance on external classifications. read the letter →

arxiv 2411.18863 v1 pith:5I7HMCX3 submitted 2024-11-28 math.GR

Michael Giudici , Luke Morgan , Cheryl E. Praeger This is my paper · ORCID

classification math.GR MSC 20D0520D0620E4520G40

keywords finitesimplegroupsp-elementsAut(T)-classesconjugacyclassesorderboundsofLietypeSingercyclesmaximalsubgroups

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

This paper proves that the order of a finite nonabelian simple group is controlled by how many automorphism-classes of elements of prime-power order it has. For a simple group $T$, define $m(T)=\max_p m_p(T)$, where $m_p(T)$ is the number of $\mathrm{Aut}(T)$-classes of elements whose order is a power of $p$. The paper shows there is an increasing function $f$ with $|T|\le f(m(T))$ for every finite nonabelian simple group $T$. Because $m(T)$ counts only $p$-power elements for a single best prime, it can be far smaller than the total number of conjugacy classes, so the result strengthens previous bounds expressed in terms of the full class count. A corollary is that every such $T$ has some prime $p$ with at least $g(|T|)$ $\mathrm{Aut}(T)$-classes of $p$-power elements; the introduction also points to consequences for relative Brauer groups of finite extensions of global fields.

What carries the argument

The load-bearing object is a cyclic subgroup $H$ (often a Singer cycle, meaning a cyclic subgroup acting regularly on the nonzero vectors of the natural module, or a maximal torus) with known normaliser structure in $\mathrm{Aut}(T)$. For a prime $s$ dividing $|H|$, let $S$ be the Sylow $s$-subgroup of $H$. Since $S$ is cyclic of order $s^b$, it contains $\varphi(|S|)=s^{b-1}(s-1)$ elements of order $|S|$, and these split into exactly $\varphi(|S|)/r$ $\mathrm{Aut}(T)$-classes, where $r=|\mathrm{N}_{\mathrm{Aut}(T)}(S):\mathrm{C}_{\mathrm{Aut}(T)}(S)|$. The hypothesis $m(T)\le n$ therefore forces $\varphi(|S|)\le rn$ and $\varphi(|S|)\mid r(n!)$, and because $r$ is controlled by the known normaliser, each prime divisor $s$ of $|H|$ is constrained. The same inequality bounds the number of such primes, and together with the divisibility identities in Lemma 2.1 this pins down $q$ and $a$. For exceptional groups the normaliser information comes from tables of maximal subgroups; for classical groups the dimension bound comes from counting unipotent classes.

What would settle it

Check the external fact the proof leans on most heavily: that the cyclic subgroup $H$ in the $E_8(q)$ row of Table 4 has normaliser $H.30$ maximal in $\mathrm{Aut}(T)$. If that normaliser failed to be maximal, the field-size bound in Proposition 5.3 would fail for $E_8(q)$; verifying it against the cited maximal-subgroup table for $E_8(q)$ would confirm the most delicate exceptional case.

Watch

Extended reading notes

Core claim

The central claim is Theorem 1.1: there exists an increasing function $f$ on the natural numbers such that, for every finite nonabelian simple group $T$, the order $|T|$ is at most $f(m(T))$, where $m(T)=\max_p m_p(T)$ and $m_p(T)$ is the number of $\mathrm{Aut}(T)$-classes of elements of $p$-power order. The proof splits the simple groups into alternating groups, classical groups of Lie type, and exceptional groups of Lie type, with the sporadic groups handled by the Monster. For alternating groups the argument uses the $\lfloor m/3\rfloor$ classes of elements of order $3$ in $\mathrm{Alt}(m)$. For classical groups it combines a lower bound of $d$ unipotent classes, where $d$ is the dimension of the natural module, with analysis of cyclic tori and their normalisers to force bounds on the field size $q$ and the exponent $a$ in $q=p^a$. For exceptional groups the proof uses cyclic subgroups $H$ whose normaliser is a maximal subgroup of $\mathrm{Aut}(T)$, listed in Tables 4 and 5, and shows that the prime divisors of $|H|$ force $q$ to be bounded.

Load-bearing premise

The proof for the exceptional families relies on previously published classifications asserting that certain cyclic subgroups have normalisers that are maximal subgroups of $\mathrm{Aut}(T)$ with exactly the listed structure; if any one of those classification facts were wrong, the field-size bound for that family would collapse.

Editorial extensions

If this is right

Corollary 1.2: there is an increasing function $g$ such that every finite nonabelian simple group $T$ has at least $g(|T|)$ $\mathrm{Aut}(T)$-classes of elements of $p$-power order for some prime $p$ dividing $|T|$.
Because $f$ is increasing, for each fixed $n$ only finitely many nonabelian simple groups have $m(T)\le n$; the theorem is thus a finiteness statement for simple groups with few $p$-element classes.
The bound generalises earlier results that used the total number of conjugacy classes, since $m(T)$ is never larger than the total class count and is often much smaller.
For simple classical groups the proof actually bounds $|T|$ by a function of $m'(T)=\max\{m_p(T),m_{S(T)\text{-exp}}(T)\}$, where the second term counts $\mathrm{Aut}(T)$-classes of elements of maximal $s$-part of the exponent for a set of primes $S(T)$.
The introduction states that Theorem 1.1 is used in follow-up work to prove a case of a conjecture about subgroups meeting every $\mathrm{Aut}(G)$-class of prime-power elements, with consequences for relative Brauer groups of finite extensions of global fields.

Reading between the lines

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

A sharper $f$ is likely: the paper's proof uses a crude divisor-counting bound to control the number of prime divisors of $|H|$, and the authors note in Remark 4.3 that a better function may be possible; replacing the $n!$-type term by an exponential bound would make the finiteness threshold practical.
The same cyclic-subgroup normaliser strategy could plausibly extend from simple groups to almost simple groups, or from $m(T)$ to the $m_{\mathrm{exp}}$ parameter for the exceptional groups, which the paper leaves open in Remark 4.4.
A computational census of $m(T)$ for simple groups of increasing order would show how fast the true optimal function grows; the theorem guarantees $m(T)\to\infty$ with $|T|$, but not whether the growth is logarithmic, polynomial, or something in between.

Share X Bluesky LinkedIn Reddit HN

Signed reviews

No signed human review yet.

Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

Short version: the theorem is new and the proof is detailed and mostly convincing. I'd send it to a referee.

The new thing is Theorem 1.1: for a finite nonabelian simple group T, |T| is bounded by a function of m(T), the maximum over primes p of the number of Aut(T)-classes of elements of p-power order. This is a real generalization of Pyber's bound, and it uses a statistic far weaker than the total number of conjugacy classes. It is exactly the kind of bound needed for Praeger's covering subgroup conjecture and for relative Brauer groups.

The proof is a genuine case analysis over alternating, classical, and exceptional groups of Lie type. Lemma 2.3 is the workhorse: it relates φ(|S|) to the number of Aut-classes of elements of order |S|, and it is applied carefully. The arithmetic in Propositions 4.2 and 5.3 is coherent, and the paper explicitly notes the n! growth of f. The literature review is fair; the claim that the single-prime version is new checks out against [15] and [28].

The soft spot is Section 5's reliance on external maximal subgroup classifications (Liebeck-Saxl-Seitz, Kleidman, Malle, Suzuki, Craven). If a row in Table 4 misstated a torus normalizer, Proposition 5.3 would fail for that family. This is a standard kind of dependence, and nothing in the rows looks suspect, but a referee should verify at least the E7 rows against Craven [7,8] and the E8 row against [24, Table 5.2]. There is also a harmless verbal overcount in Lemma 4.1: for PΩ^-(2d,q) with d even, the count should be d−1 nonidentity classes, but the identity supplies the missing class, so the lemma still holds.

The stress-test note's assessment matches my reading: no structural flaw, and the dependence on the literature is ordinary.

This paper is for finite group theorists working on Landau-type bounds, and for number theorists using class-count arguments. It deserves a serious referee.

Referee Report

0 major / 7 minor

Summary. The paper proves that there exists an increasing function f such that every finite nonabelian simple group T satisfies |T| ≤ f(m(T)), where m(T) is the maximum, over all primes p, of the number of Aut(T)-classes of elements of p-power order in T. The proof combines an elementary counting lemma (Lemma 2.3), which controls the number of Aut(T)-classes meeting a cyclic subgroup in terms of the index of its normalizer, with a case-by-case analysis over the finite simple groups: alternating groups, classical groups of Lie type, and exceptional groups of Lie type. For classical groups the argument uses lower bounds on unipotent class numbers and carefully chosen cyclic tori whose normalizers have small index; for exceptional groups it relies on known maximal-subgroup classifications to identify cyclic subgroups with maximal normalizers. The paper also gives a corollary lower bound on the number of Aut(T)-classes of p-elements in terms of |T| and discusses applications to relative Brauer groups.

Significance. If correct, this is a substantial strengthening of previous results of Pyber and of Hethelyi and Kulshammer, replacing bounds on the total number of conjugacy classes by a much weaker statistic, the maximum over primes of the number of Aut(T)-classes of p-power elements. The proof is structured and auditable: the counting arguments are elementary, the constants are explicit, and there are no fitted parameters. The main risk is the external dependence of Section 5: Lemma 5.1 and Table 4 rest on the maximal-subgroup classifications in [7,8,20,21,24,26,35], and a misstated normalizer row would break Proposition 5.3 for the corresponding family. This is a standard kind of reliance in the subject, and I found no internal inconsistency or circularity; the use of the classification of finite simple groups is independent of the theorem being proved.

minor comments (7)

[§5, Proposition 5.3, Case a)i] The sentence beginning 'Since T is pp-bounded by n, it follows from Lemma 2.3 that |C(s)| divides z|Out(T)|n!' misstates the conclusion of Lemma 2.3: what divides z|Out(T)|n! is φ(|Hs|), not |C(s)|. The deduction that s−1 divides 360a(n!) uses φ(|Hs|)=r|C(s)| with r dividing z|Out(T)|, and the current wording is insufficient.
[§5, Proposition 5.3, Case a)ii] The estimate '|Hs| ≤ z|Out(T)|n' for odd s drops the factor s/(s−1), which is explicitly included in the corresponding step of Case a)i). The correct bound is |Hs| ≤ 2z|Out(T)|n ≤ 8an; the final qualitative conclusion is unaffected once this factor is restored.
[§5, Lemma 5.1] In the proof of statement (1), the one-line deduction 'Since s is coprime to z, it follows that Hs = Os(NT(H))' deserves a brief justification when s divides |D|: because D is simple nonabelian, Os(D)=1, and because Z has order coprime to s, the largest normal s-subgroup of NT(H) lies in D×H and equals Hs. Please add this clarification.
[§4, Proposition 4.2, linear exceptional case] In the displayed chain 'q < (q^3−1)/(q−1)^2 ≤ Prod(3,q)/(q−1)_3 = 3|H|', the final denominator is the 3-part of q−1, not the cube of q−1; using the same subscript notation as elsewhere would avoid ambiguity.
[§5, Proposition 5.3, Case b] The sentence 'Let H be the cyclic subgroup of T defined in Lemma 5.1' should refer to Lemma 5.2, since this case treats the groups in Table 5.
[§5, Proposition 5.3, opening reduction] The phrase 'we may assume q > 2 so that we may use all rows of Table 4' should explicitly say that q=2 gives only finitely many exceptional groups and that their orders can be absorbed into the function h; as written the reduction is slightly too quick.
[§3 and §1] There are several small typos: 'elements of elements order 3' in Lemma 3.1, 'for for all positive integers m' in Remark 3.2, and 'elements of p-elements' in the definition of mp(T) in the Introduction.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the derivation is a genuine upper bound on |T| in terms of m(T), with all load-bearing external inputs independently cited.

full rationale

The paper's central claim is that |T| is bounded by an increasing function of m(T), where m(T) counts Aut(T)-classes of p-power-order elements. The proof never defines m(T) in terms of |T|, fits a parameter and then 'predicts' a related quantity, or imports a uniqueness theorem from the authors' prior work. The core chain is: Lemma 2.3 derives divisibility constraints on phi(|S|) from the assumption m(T) <= n; Lemma 4.1 shows a classical group has at least d unipotent classes, so d <= n; Proposition 4.2 uses cyclic torus subgroups and Sylow-subgroup counts to bound q and hence |T|; Section 5 does the same for exceptional groups using cyclic subgroups whose normalizers are known maximal subgroups. The only self-citations are [12], a future application of Theorem 1.1 and not used in the proof; [13], an elementary 2-adic valuation lemma used in Lemma 2.1(iii); and [29], the standard normalizer structure of a Singer cycle in GL(1,q^m). These are independent published facts, not unverified assertions of the target theorem. The main external dependence is the maximal-subgroup classification literature in Section 5 (Liebeck-Saxl-Seitz, Kleidman, Malle, Suzuki, Craven), but that is normal external support, and no cited result is shown to assume the conclusion of Theorem 1.1. No fitted inputs, no self-definitional equations, and no renamings of known results occur. The proof is therefore not circular.

Assumptions & free parameters 0 free parameters · 6 assumptions · 0 invented entities

The central claim rests on these external classification and number-theoretic results. No free parameters or invented entities are introduced. The paper is conditional on the accepted literature of finite simple groups and their subgroups.

assumptions (6)

domain assumption The Classification of Finite Simple Groups: every finite nonabelian simple group is alternating, classical, exceptional of Lie type, or sporadic.
Invoked in Section 6 to reduce Theorem 1.1 to the four families treated in Sections 3, 4, and 5. It is a standard theorem in the field, not proved in this paper.
domain assumption Maximal subgroup classifications for exceptional groups: for each group in Table 4, there exists a cyclic subgroup H with the listed order such that N_{Aut(T)}(H) is maximal in Aut(T), with normalizer structure as in the table.
Used in Lemma 5.1 and Proposition 5.3 Case a. The paper cites [24], [35], [21], [26], [20], [8], [7] for these facts. If any classification is wrong, the proof fails for that family.
domain assumption For each group in Table 5, there exists a cyclic subgroup H with the listed order such that N_T(H) is contained in a unique maximal subgroup M with the structure given.
Used in Lemma 5.2 and Proposition 5.3 Case b for G2(q) and F4(q) with q odd. Cites [6], [21], [24], [8].
domain assumption Classification of unipotent conjugacy classes in finite classical groups: for each family in Table 1, there are at least d Aut(T)-classes of unipotent elements.
Used in Lemma 4.1 to bound d ≤ n. The paper cites [1,5,9,25,36] and states the classes are well known.
standard math The divisor bound: for a positive integer N, the number of positive divisors of N is less than 2√N.
Used in Propositions 4.2 and 5.3 to bound the number of primes in S(m) or S(2m) and in π(H). Cited to [31, Section 8.3].
standard math Bertrand's postulate: for every integer d > 1 there is a prime m with d/2 < m ≤ d.
Used at the start of Proposition 4.2 to choose a prime m.

how reviews work

0 comments

Cite this review

Pith. "Pith review of Finite simple groups have many classes of $p$-elements." pith.science (2026). https://pith.science/paper/5I7HMCX3

@misc{pith2026241118863,
  author       = {Pith},
  title        = {Pith review of: Finite simple groups have many classes of $p$-elements},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/5I7HMCX3}},
  note         = {Machine review of arXiv:2411.18863}
}

abstract

For an element $x$ of a finite group $T$, the $\mathrm{Aut}(T)$-class of $x$ is the set $\{ x^\sigma\mid \sigma\in \mathrm{Aut}(T)\}$. We prove that the order $|T|$ of a finite nonabelian simple group $T$ is bounded above by a function of the parameter $m(T)$, where $m(T)$ is the maximum, over all primes $p$, of the number of $\mathrm{Aut}(T)$-classes of elements of $T$ of $p$-power order. This bound is a substantial generalisation of results of Pyber, and of H\'ethelyi and K\"ulshammer, and it has implications for relative Brauer groups of finite extensions of global fields.

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

36 extracted references · 35 canonical work pages

[15]

H´ ethelyi and B

L. H´ ethelyi and B. K¨ ulshammer, Elements of prime power order and their conjugacy classes in ﬁnite groups, J. Aust. Math. Soc. 78 (2005) 291–295

work page 2005
[28]

Moret´ o, and H

A. Moret´ o, and H. N. Nguyen, Variations of Landau’s theorem for p-regular and p-singular conjugacy classes. Israel J. Math. 212 (2016), 961–987

work page 2016
[1]

Aschbacher and G

M. Aschbacher and G. M. Seitz, Involutions in Chevalley groups ov er ﬁelds of even order. Nagoya Math. J., 63 (1976), 1–91

work page 1976
[2]

Baumeister, A Mar´ oti, and H

B. Baumeister, A Mar´ oti, and H. P. Tong-Viet, Finite groups hav e more conjugacy classes. Forum Math. 29 (2017), 259–275

work page 2017
[3]

Brauer, Representations of ﬁnite groups, in Lectures on modern mathematics, Vol

R. Brauer, Representations of ﬁnite groups, in Lectures on modern mathematics, Vol. I (ed. T. L. Saaty), Wiley, New York, 1963

work page 1963
[4]

T. C. Burness, Fixed point ratios in actions of ﬁnite classical grou ps, II. J. Algebra , 309 (2007) 80–138

work page 2007
[5]

T. C. Burness, and M. Giudici, Classical Groups, Derangements and Primes Australian Mathematical Society Lecture Series, No. 25 Cambridge University Press, 2015

work page 2015
[6]

B. N. Cooperstein. Maximal Subgroups of G2(2n). J. Algebra, 70 (1981), 23–36

work page 1981

Show all 36 references

[7]

D. A. Craven, The maximal subgroups of E7(q) and related almost simple groups, (2022), arXiv:2201.07081

2022 arXiv
[8]

D. A. Craven, The maximal subgroups of the exceptional group s F4(q), E6(q) and 2E6(q) and related almost simple groups, Invent. Math , 234 (2023), 637–719

2023
[9]

De Franceschi and M

G. De Franceschi and M. W. Liebeck and E. A. O’Brien, Conjugacy in ﬁnite classical groups, (2024) arXiv:2401.07557

2024 arXiv
[10]

B. Fein, W. M. Kantor and M. Schacher, Relative Brauer groups . II., J. Reine Angew. Math. 328 (1981), 39–57

1981
[11]

Gao and Y

Y. Gao and Y. Yang, Lower bound for the number of conjugacy classes and irreducible characters, Monatsh. Math. 203 (2024), 341–355. 16

2024
[12]

Giudici, L

M. Giudici, L. Morgan and C. E. Praeger, Simple group orders and prime power coverings of groups, Preprint (2024)

2024
[13]

Guest and C

S. Guest and C. E. Praeger, Proportions of elements with given 2-part order in ﬁnite classical groups of odd characteristic. J. Algebra 372 (2012), 637–660

2012
[14]

R. M. Guralnick, Zeros of permutation characters with applicat ions to prime splitting and Brauer groups, J. Algebra 131 (1990), 294–302

1990
[16]

N. N. Hung, T. M. Keller and Y. Yang, A lower bound for the numbe r of odd-degree representations of a ﬁnite group. Math. Z. 298 (2021), 1559–1572

2021
[17]

Huppert, Endliche Gruppen

B. Huppert, Endliche Gruppen. I , Die Grundlehren der Mathematischen Wissenschaften, Band 134, Springer-Verlag, Berlin-New York, 1967

1967
[18]

Jaikin-Zapirain, On the number of conjugacy classes of ﬁnite nilpotent groups

A. Jaikin-Zapirain, On the number of conjugacy classes of ﬁnite nilpotent groups. Adv. Math. 227 (2011), 1129–1143

2011
[19]

T. M. Keller, Finite groups have even more conjugacy classes, Israel J. Math. 181 (2011), 433–444

2011
[20]

P. B. Kleidman, The maximal subgroups of the Steinberg triality g roups 3D4(q) and of their automorphism groups, J. Algebra 115 (1988), 182–199

1988
[21]

P. B. Kleidman, The maximal subgroups of the Chevalley groups G2(q) with q odd, the Ree groups 2G2(q), and their automorphism groups, J. Algebra 117 (1988), 30–71

1988
[22]

Kleidman, M

P. Kleidman, M. Liebeck, The subgroup structure of the ﬁnite classical groups , London Mathematical Society Lecture Note Series, 129, Cambridge University Press, Cambridge, (1990)

1990
[23]

Landau, ¨Uber die klassenzahl der bin¨ aren quadratischen formen von nega tiver discrimi- nante, Math

E. Landau, ¨Uber die klassenzahl der bin¨ aren quadratischen formen von nega tiver discrimi- nante, Math. Ann. 56 (1903), 671–676

1903
[24]

M. W. Liebeck, J. Saxl and G. M. Seitz, Subgroups of maximal ra nk in ﬁnite exceptional groups of Lie type, Proc. London Math. Soc. (3) 65 (1992), 297–325

1992
[25]

M. W. Liebeck, and G. M. Seitz, Unipotent and nilpotent classes in simple algebraic groups and Lie algebras . Math. Surveys Monogr., vol. 180. Amer. Math. Soc., Providence, RI., 2012

2012
[26]

G. Malle. The maximal subgroups of 2F4(q2). J. Algebra, 139, 52–69, (1991)

1991
[27]

Malle, G

G. Malle, G. Navarro, J. B. Olsson, Zeros of characters of ﬁnit e groups, J. Group Theory 3 (2000), 353—368

2000
[29]

P. M. Neumann and C. E. Praeger. A recognition algorithm for sp ecial linear groups, Proc. London Math. Soc.(3) 65 (1992), 555–603

1992
[30]

A. C. Niemeyer and C. E. Praeger. Estimating proportions of ele ments in ﬁnite groups of Lie type, J. Algebra 324 (2010), 122–145

2010
[31]

Niven, H

I. Niven, H. S. Zuckerman and H. L. Montgomery, An introduction to the theory of numbers , Wiley, New York, Fifth Edition, 1991. 17

1991
[32]

C. E. Praeger, Kronecker classes of ﬁelds and covering subgr oups of ﬁnite groups, J. Aust. Math. Soc. 57 (1994), 17–34

1994
[33]

Pyber, Finite groups have many conjugacy classes, J

L. Pyber, Finite groups have many conjugacy classes, J. London Math. Soc. (2) 46 (1992), 239–249

1992
[34]

Ribenboim, The little book of bigger primes

P. Ribenboim, The little book of bigger primes. Second edition, Springer-Verlag, New York, 2004

2004
[35]

M. Suzuki. On a class of doubly transitive groups. Ann. Math. Vol. 75. No. 1. (1962), 105– 145

1962
[36]

G. E. Wall, On the conjugacy classes in the unitary, symplectic an d orthogonal groups. J. Aust. Math. Soc. 3, (1963), 1–62. 18

1963

Pith tools

Reviewed August 12, 2026 · model on record in the stance chip above.

[1] [15]

H´ ethelyi and B

L. H´ ethelyi and B. K¨ ulshammer, Elements of prime power order and their conjugacy classes in ﬁnite groups, J. Aust. Math. Soc. 78 (2005) 291–295

work page 2005

[2] [28]

Moret´ o, and H

A. Moret´ o, and H. N. Nguyen, Variations of Landau’s theorem for p-regular and p-singular conjugacy classes. Israel J. Math. 212 (2016), 961–987

work page 2016

[3] [1]

Aschbacher and G

M. Aschbacher and G. M. Seitz, Involutions in Chevalley groups ov er ﬁelds of even order. Nagoya Math. J., 63 (1976), 1–91

work page 1976

[4] [2]

Baumeister, A Mar´ oti, and H

B. Baumeister, A Mar´ oti, and H. P. Tong-Viet, Finite groups hav e more conjugacy classes. Forum Math. 29 (2017), 259–275

work page 2017

[5] [3]

Brauer, Representations of ﬁnite groups, in Lectures on modern mathematics, Vol

R. Brauer, Representations of ﬁnite groups, in Lectures on modern mathematics, Vol. I (ed. T. L. Saaty), Wiley, New York, 1963

work page 1963

[6] [4]

T. C. Burness, Fixed point ratios in actions of ﬁnite classical grou ps, II. J. Algebra , 309 (2007) 80–138

work page 2007

[7] [5]

T. C. Burness, and M. Giudici, Classical Groups, Derangements and Primes Australian Mathematical Society Lecture Series, No. 25 Cambridge University Press, 2015

work page 2015

[8] [6]

B. N. Cooperstein. Maximal Subgroups of G2(2n). J. Algebra, 70 (1981), 23–36

work page 1981