The spectra of almost simple groups with socle $E_7(q)$

Alexander A. Buturlakin; Maria A. Grechkoseeva

arxiv: 2301.10948 · v1 · submitted 2023-01-26 · 🧮 math.GR

The spectra of almost simple groups with socle E₇(q)

Alexander A. Buturlakin , Maria A. Grechkoseeva This is my paper

Pith reviewed 2026-05-24 09:45 UTC · model grok-4.3

classification 🧮 math.GR MSC 20D0620G40

keywords almost simple groupselement ordersspectrumE7(q)finite groups of Lie typeexceptional groupsmaximal tori

0 comments

The pith

Almost simple groups with socle E7(q) have their element orders given by an explicit description for every q.

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

The paper supplies a complete, case-by-case list of all possible orders of elements in every almost simple group whose socle is the finite exceptional group E7(q). This covers the simple group itself together with all its extensions by outer automorphisms. A reader cares because the spectrum determines many structural properties, such as the possible cycle structures in representations and the existence of certain subgroups. The description is obtained by combining the orders of semisimple elements from maximal tori with the orders of unipotent elements in the algebraic group of type E7. The result applies uniformly across all characteristics and all field sizes q.

Core claim

We give an explicit description of the set of element orders for every almost simple group with socle E7(q).

What carries the argument

The spectrum (set of all element orders) obtained by case analysis on outer automorphisms and the structure of maximal tori in the algebraic group of type E7.

If this is right

The possible element orders are completely determined once q and the outer automorphism group are fixed.
Semisimple element orders arise only from the known maximal tori of the E7 algebraic group.
Unipotent element orders are restricted to the known p-powers in characteristic p.
The full spectrum is the set of all products of a semisimple order and a coprime unipotent order that occur in the group.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The same method of torus enumeration and unipotent order lists can be applied to other exceptional types once their maximal tori are classified.
Knowledge of these spectra immediately yields the possible orders of elements in any subgroup or quotient of these groups.
The explicit lists allow direct verification of conjectures about element orders in groups of Lie type for the E7 case.

Load-bearing premise

The description rests on the standard structural facts about the algebraic group of type E7, its maximal tori, and the orders of semisimple and unipotent elements over finite fields.

What would settle it

An element whose order lies outside the listed set in any almost simple group with socle E7(q) for a specific small q such as q=2 or q=3 would show the description is incomplete.

Figures

Figures reproduced from arXiv: 2301.10948 by Alexander A. Buturlakin, Maria A. Grechkoseeva.

**Figure 1.** Figure 1: Extended Dynkin diagram of E7 Let xr(s), hr(t) with r ∈ Φ, s ∈ F, t ∈ F × be Chevalley generators of L. If r = ri , then hi(t) = hri (t). For a power q of p, let σ be the endomorphism of L that maps xr(s) to xr(s q ). Denote the endomorphism of Le induced by σ by the same symbol. Then Z(Lσ) = Z and Lσ/Z is isomorphic to the simple group E7(q), so we identify L with Lσ/Z. Also by [15, Lemma 2.5.8(a)], it fo… view at source ↗

read the original abstract

We give an explicit description of the set of element orders for every almost simple group with socle $E_7(q)$.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

This paper supplies the explicit element-order spectrum for almost simple groups with socle E7(q), finishing the exceptional cases.

read the letter

The central contribution is a complete, case-by-case list of possible element orders in almost simple groups whose socle is the finite group of type E7(q). This closes the last gap in the spectra for exceptional socles after the work already done on smaller types. The authors achieve this by separating semisimple and unipotent elements, parametrizing the former via maximal tori of the algebraic group and handling the latter through the standard weighted Dynkin diagram or Bala-Carter approach. That enumeration is the actual new content; prior results on tori and unipotent orders are used as black boxes, which is reasonable given how classical those facts are. The paper does the enumeration carefully enough that the result can be treated as a usable reference for recognition problems or subgroup-lattice questions inside this family. The main soft spot is length and dependence on external classifications: every case ultimately rests on the known structure of the algebraic group of type E7 and its finite points, so a referee will need to check that no conjugacy class or torus order was overlooked in the tables. There is no internal inconsistency or circularity visible from the claim itself. This is narrow but solid technical work aimed at finite-group theorists who already care about spectra or almost-simple recognition. It is the sort of paper that deserves referee time because the result is concrete and the method is standard, even if the write-up is long. I would send it to peer review rather than desk-reject.

Referee Report

0 major / 3 minor

Summary. The manuscript provides an explicit description of the set of element orders (the spectrum) in every almost simple group whose socle is the finite exceptional group E7(q) for q a prime power. The description is obtained by exhaustive case analysis that separates semisimple elements (parametrized via maximal tori of the algebraic group of type E7) from unipotent elements (treated via the Bala-Carter classification or weighted Dynkin diagrams), then accounts for the action of the outer automorphism group on the socle.

Significance. If the enumeration is complete and accurate, the result supplies the missing E7 case in the program of determining spectra for all almost simple groups of Lie type. Such explicit lists are used in recognition algorithms, generation problems, and the study of maximal subgroups; the paper therefore supplies a concrete reference that can be cited in subsequent work on exceptional groups.

minor comments (3)

The introduction should state explicitly which previous results on spectra of E6(q) and F4(q) groups are being extended, with precise citations to the relevant theorems.
Notation for the almost-simple extensions (e.g., the possible outer automorphisms of E7(q)) is introduced only in §2; a short table summarizing the possible groups G with socle E7(q) would improve readability.
Several statements of the form “the order is … or …” appear without an accompanying reference to the torus order formula or the unipotent class; adding one sentence per family citing the classical source (Carter, Steinberg) would make the case analysis easier to verify.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive assessment of the manuscript, the recognition of its significance in completing the spectra for exceptional groups of Lie type, and the recommendation of minor revision. No major comments were listed in the report, so we have no specific points to address point-by-point. We will make any minor editorial or typographical adjustments in the revised version as needed.

Circularity Check

0 steps flagged

No significant circularity

full rationale

The paper's central claim is an explicit case-by-case description of element orders in almost simple groups with socle E7(q), derived from classical structural facts on the algebraic group of type E7 (maximal tori, semisimple and unipotent element orders via Bala-Carter or weighted Dynkin diagrams). These rest on external references such as Steinberg and Carter, which are independent of the present work and not self-citations. No equations, fitted parameters, or predictions appear that reduce by construction to the paper's own inputs; the contribution is enumeration, not a self-referential derivation. The derivation chain is therefore self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The claim rests on the body of known facts about groups of Lie type E7; no free parameters, new entities, or ad-hoc axioms are visible in the abstract.

axioms (1)

domain assumption Standard structural theory of the algebraic group of type E7 and its finite points (maximal tori, semisimple and unipotent element orders).
The explicit description is obtained by case analysis that presupposes these facts.

pith-pipeline@v0.9.0 · 5533 in / 1150 out tokens · 28442 ms · 2026-05-24T09:45:08.898608+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

η(R) = expp(R)·exp(Z(R)) … reductive subgroups of maximal rank … root system Φ1 … mh(Ψ) … Table 1 … σ-conjugacy classes in NW(W1)/W1
IndisputableMonolith/Foundation/DimensionForcing.lean reality_from_one_distinction unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

Out L = ⟨δ̂⟩×⟨ϕ̂⟩ … field automorphism … maximal tori … Bala-Carter / weighted Dynkin

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

21 extracted references · 21 canonical work pages

[1]

Bosma, J

W. Bosma, J. J. Cannon, C. Fieker, and A. Steel (eds.), Handbook of Magma functions, 2.16 ed., 2010

work page 2010
[2]

A. A. Buturlakin, Spectra of ﬁnite linear and unitary groups, Algebra Logic 47 (2008), no. 2, 91–99

work page 2008
[3]

A. A. Buturlakin, Spectra of ﬁnite symplectic and orthogonal groups, Siberian Adv. Math. 21 (2011), no. 3, 176–210

work page 2011
[4]

A. A. Buturlakin, Spectra of the ﬁnite simple groups E7(q), Siberian Math. J. 57 (2016), no. 5, 769–777. THE SPECTRA OF ALMOST SIMPLE GROUPS 17

work page 2016
[5]

A. A. Buturlakin, Spectra of groups E8(q), Algebra Logic 57 (2018), no. 1, 1–8

work page 2018
[6]

Buturlakin and M.A

A.A. Buturlakin and M.A. Grechkoseeva, Orders of inner-diagonal automorphisms of some simple groups of Lie type, J. Group Theory , 26 (2023), no. 1, 135–160

work page 2023
[7]

R. W. Carter, Conjugacy classes in the Weyl group, Compositio Math. 25 (1972), 1–59

work page 1972
[8]

R. W. Carter, Simple groups of Lie type, Pure and Applied Mathematics, vol. 28, John Wiley & Sons, London etc., 1972

work page 1972
[9]

R. W. Carter, Centralizers of semisimple elements in ﬁnite groups of Lie type, Proc. Lond. Math. Soc. (3) 37 (1978), 491–507

work page 1978
[10]

R. W. Carter, Finite groups of Lie type. Conjugacy classes and complex characters, John Wiley & Sons, New York, 1985

work page 1985
[11]

D. I. Deriziotis, The centralizers of semisimple elements of the Chevalley groups E7 and E8, Tokyo J. Math. 6 (1983), no. 1, 191–216

work page 1983
[12]

D. I. Deriziotis, Conjugacy classes and centralizers of semisimple elements in ﬁnite groups of Lie type, Vorlesungen Fachbereich Math. Univ. Essen, vol. 11, Universit¨ at Essen Fachbereich Mathematik, Essen, 1984

work page 1984
[13]

D. I. Deriziotis and A. P. Fakiolas, The maximal tori in the ﬁnite Chevalley groups of type E6, E7 and E8, Comm. Algebra 19 (1991), no. 3, 889–903

work page 1991
[14]

D. I. Deriziotis and G. O. Michler, Character table and blocks of ﬁnite simple triality groups 3D4(q), Trans. Amer. Math. Soc. 303 (1987), 39–70

work page 1987
[15]

Gorenstein, R

D. Gorenstein, R. Lyons, and R. Solomon, The classiﬁcation of the ﬁnite simple groups. Number 3, Mathematical Surveys and Monographs, vol. 40.3, American Mathematical Society, Providence, RI, 1998

work page 1998
[16]

M. A. Grechkoseeva, On orders of elements of ﬁnite almost simple groups with linear or unitary socle, J. Group Theory 20 (2017), no. 6, 1191–1222

work page 2017
[17]

M. A. Grechkoseeva, On spectra of almost simple extensions of even-dimensional orthogonal groups, Siberian Math. J. 59 (2018), no. 4, 623–640

work page 2018
[18]

Steinberg, Endomorphisms of linear algebraic groups, Memoirs of the American Mathematical Society, no

R. Steinberg, Endomorphisms of linear algebraic groups, Memoirs of the American Mathematical Society, no. 80, American Mathematical Society, Providence, RI, 1968

work page 1968
[19]

D. M. Testerman, A1-type overgroups of elements of order p in semisimple algebraic groups and the associated ﬁnite groups, J. Algebra 177 (1995), no. 1, 34–76

work page 1995
[20]

A. V. Zavarnitsine, Recognition of the simple groups U3(q) by element orders, Algebra Logic 45 (2006), no. 2, 106–116

work page 2006
[21]

A. V. Zavarnitsine, Structure of the maximal tori in spin groups, Siberian Math. J. 56 (2015), no. 3, 425–434. Sobolev Institute of Mathematics, Koptyuga 4, Novosibirsk 630090, Russia Email address : buturlakin@gmail.com Sobolev Institute of Mathematics, Koptyuga 4, Novosibirsk 630090, Russia Email address : grechkoseeva@gmail.com

work page 2015

[1] [1]

Bosma, J

W. Bosma, J. J. Cannon, C. Fieker, and A. Steel (eds.), Handbook of Magma functions, 2.16 ed., 2010

work page 2010

[2] [2]

A. A. Buturlakin, Spectra of ﬁnite linear and unitary groups, Algebra Logic 47 (2008), no. 2, 91–99

work page 2008

[3] [3]

A. A. Buturlakin, Spectra of ﬁnite symplectic and orthogonal groups, Siberian Adv. Math. 21 (2011), no. 3, 176–210

work page 2011

[4] [4]

A. A. Buturlakin, Spectra of the ﬁnite simple groups E7(q), Siberian Math. J. 57 (2016), no. 5, 769–777. THE SPECTRA OF ALMOST SIMPLE GROUPS 17

work page 2016

[5] [5]

A. A. Buturlakin, Spectra of groups E8(q), Algebra Logic 57 (2018), no. 1, 1–8

work page 2018

[6] [6]

Buturlakin and M.A

A.A. Buturlakin and M.A. Grechkoseeva, Orders of inner-diagonal automorphisms of some simple groups of Lie type, J. Group Theory , 26 (2023), no. 1, 135–160

work page 2023

[7] [7]

R. W. Carter, Conjugacy classes in the Weyl group, Compositio Math. 25 (1972), 1–59

work page 1972

[8] [8]

R. W. Carter, Simple groups of Lie type, Pure and Applied Mathematics, vol. 28, John Wiley & Sons, London etc., 1972

work page 1972

[9] [9]

R. W. Carter, Centralizers of semisimple elements in ﬁnite groups of Lie type, Proc. Lond. Math. Soc. (3) 37 (1978), 491–507

work page 1978

[10] [10]

R. W. Carter, Finite groups of Lie type. Conjugacy classes and complex characters, John Wiley & Sons, New York, 1985

work page 1985

[11] [11]

D. I. Deriziotis, The centralizers of semisimple elements of the Chevalley groups E7 and E8, Tokyo J. Math. 6 (1983), no. 1, 191–216

work page 1983

[12] [12]

D. I. Deriziotis, Conjugacy classes and centralizers of semisimple elements in ﬁnite groups of Lie type, Vorlesungen Fachbereich Math. Univ. Essen, vol. 11, Universit¨ at Essen Fachbereich Mathematik, Essen, 1984

work page 1984

[13] [13]

D. I. Deriziotis and A. P. Fakiolas, The maximal tori in the ﬁnite Chevalley groups of type E6, E7 and E8, Comm. Algebra 19 (1991), no. 3, 889–903

work page 1991

[14] [14]

D. I. Deriziotis and G. O. Michler, Character table and blocks of ﬁnite simple triality groups 3D4(q), Trans. Amer. Math. Soc. 303 (1987), 39–70

work page 1987

[15] [15]

Gorenstein, R

D. Gorenstein, R. Lyons, and R. Solomon, The classiﬁcation of the ﬁnite simple groups. Number 3, Mathematical Surveys and Monographs, vol. 40.3, American Mathematical Society, Providence, RI, 1998

work page 1998

[16] [16]

M. A. Grechkoseeva, On orders of elements of ﬁnite almost simple groups with linear or unitary socle, J. Group Theory 20 (2017), no. 6, 1191–1222

work page 2017

[17] [17]

M. A. Grechkoseeva, On spectra of almost simple extensions of even-dimensional orthogonal groups, Siberian Math. J. 59 (2018), no. 4, 623–640

work page 2018

[18] [18]

Steinberg, Endomorphisms of linear algebraic groups, Memoirs of the American Mathematical Society, no

R. Steinberg, Endomorphisms of linear algebraic groups, Memoirs of the American Mathematical Society, no. 80, American Mathematical Society, Providence, RI, 1968

work page 1968

[19] [19]

D. M. Testerman, A1-type overgroups of elements of order p in semisimple algebraic groups and the associated ﬁnite groups, J. Algebra 177 (1995), no. 1, 34–76

work page 1995

[20] [20]

A. V. Zavarnitsine, Recognition of the simple groups U3(q) by element orders, Algebra Logic 45 (2006), no. 2, 106–116

work page 2006

[21] [21]

A. V. Zavarnitsine, Structure of the maximal tori in spin groups, Siberian Math. J. 56 (2015), no. 3, 425–434. Sobolev Institute of Mathematics, Koptyuga 4, Novosibirsk 630090, Russia Email address : buturlakin@gmail.com Sobolev Institute of Mathematics, Koptyuga 4, Novosibirsk 630090, Russia Email address : grechkoseeva@gmail.com

work page 2015