$\Text {M} $, $\Text {B} $ and $\Text {Co} 1 $ Are Recognisable by Their

$$\Text {M} $, $\Text {B} $ and $\Text {Co} 1 $ Are Recognisable by Their$

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G Recognisable? Γ(H)=Γ(G)? Reference

Co2,J1, M22, M23, M24 Yes Hagie [4] M11 2-recognisable H =L2(11) Hagie [4] J2, M12 Unrecognisable Hagie [4] J4 Yes Zavarnitsine [10] Ru Yes Kondrat’ev [6] HN 2-recognisable H = HN.2 Kondrat’ev [6] Fi22 3-recognisable H Fi22.2, Suz.2 Kondrat’ev [6] ∈{ } He, McL, Co3 Unrecognisable Kondrat’ev [6] ′ Fi23, Fi24,J3, Ly, O’N, Suz, Th Yes Kondrat’ev [7] HS 2-recognisable H = U6(2) Kondrat’ev [7] M, B, Co1 Yes Theorem1.1 Table 1. Recognisability of the sporadic simple groups by their prime graphs.

freely on Op(G), and hence on a minimal normal subgroup of Op(G), which is necessarily elementary abelian and can therefore be regarded as an FCo1-module for some field F of characteristic p. (Otherwise, the product of g with a commuting h Op(G) would have ∈ order 23p, so Γ(G) would contain the edge 23,p , contradicting Γ(G) = Γ(Co1).) { } In particular, every element of order 23 in a maximal subgroup Co2 of G/F (G) ∼= Co1 must act fixed-point freely on O3(G). However, according to the 3-modular Brauer character table of Co2, which is available in GAP [1, 5], the elements of order 23 in Co2 do not act fixed-point freely on any irreducible FG-module defined over a field F of characteristic 3. Therefore, O3(G) must be trivial, and so O2(G) must be nontrivial. Let V be a minimal normal subgroup of O2(G), and let χ be the corresponding Brauer character for Co1. Consider again a maximal subgroup Co2. According to the 2-modular Brauer character table of Co2, the only irreducible module for Co2 in characteristic 2 on which elements of order 23 act fixed-point freely is the module, call it W , of dimension 22. Therefore, each of the k composition factors of the restriction V Co2 of V to Co2 ↓ is isomorphic to W , and so the restriction of χ to Co2 is k times the Brauer character of W . Now consider a maximal subgroup 3.Suz.2 of Co1. There are exactly 6 non-trivial conjugacy classes of elements of odd order in Co1 that intersect both 3.Suz.2 and Co2. These classes are listed in Table 2, together with the corresponding values of the hypothesised character χ, and the corresponding classes in 3.Suz.2 and Co2. The restriction of χ to 3.Suz.2 must be a linear combination of the irreducible 2-modular Brauer characters for 3.Suz.2, with non-negative integer coefficients. Using the 2-modular Brauer character table for 3.Suz.2, we thereby obtain a system of equations for the multiplicities of the characters appearing in this restriction. There are 9 equations (one for each conjugacy class of 3.Suz.2 in Table 2, plus one for the identity element) in 27 variables (one for each irreducible 2-modular Brauer character). Using Mathematica [9], we check that this system has no (non-trivial) non-negative integer solutions, so we have a contradiction. Therefore, O2(G) is also trivial, so F (G) is trivial and hence G ∼= Co1, as claimed.

3. Proof of Theorem 1.1 — M

Recall that M = 246 320 59 76 112 133 17 19 23 29 31 41 47 59 71. The prime graph Γ(M)| | has three· isolated· · · vertices· 41· , · 59· and· 71· ,· and· all other· · vertices { } { } { } lie in a connected component. Our argument is similar to the argument given for Co1. Suppose that G is a ﬁnite group with Γ(G) = Γ(M). Then [4, Theorem 3] implies that G/F (G) ∼= M and that F (G) is a 3-group. If O3(G) is nontrivial, then every element of order 71 in G must act ﬁxed-point freely on O3(G). In particular, every element of order M, B AND Co1 ARERECOGNISABLEBYTHEIRPRIMEGRAPHS 3

Co1 Co2 3.Suz.2 χ 3B 3B 3C, 3D 4k 3C 3A 3E 5k 5B 5B 5B− 2k 9C 9A 9B k 11A 11A 11A 0 15D 15A 15E k − Table 2. The 2-regular conjugacy classes of Co1 that intersect both maximal subgroups Co2 and 3.Suz.2, and the corresponding conjugacy classes in these subgroups. Class names are as in the ATLAS [3]. The ﬁnal column lists the corresponding values of the 2-modular Brauer character χ on the hypothesised Co1-module V in the proof of Theorem 1.1.

71 in a maximal subgroup L2(71) of G/F (G) ∼= M must act fixed-point freely on O3(G). Let V be a minimal normal subgroup of O3(G), and let χ be the corresponding Brauer character for M. By the 3-modular Brauer character table of L2(71), the only irreducible modules for L2(71) in characteristic 3 on which elements of order 71 act fixed-point freely are the two modules of dimension 70 and the two modules of dimension 35. Hence, every composition factor of V L2(71) must be isomorphic to one of these four modules. The ↓ conjugacy classes 2B and 7B of M intersect L2(71) in its conjugacy classes 2A and 7A–C. The elements of the L2(71)-classes 7A–C all have vanishing Brauer character on each of the four aforementioned modules, so we infer that χ(7B) = 0. We now use the fact that every element of order 41 in G must act fixed-point freely on V . In particular, every such element in a maximal subgroup L2(41) must act fixed-point freely on V . (Recall that such a subgroup exists [8].) The only irreducible modules for L2(41) in characteristic 3 on which elements of order 41 act fixed-point freely are the four modules of dimension 40. Each of the corresponding Brauer characters vanishes on the unique class of involutions in L2(41). The elements of this class belong to the class 2B in M, so we infer 1+2 that χ(2B) = 0. Finally, we consider a subgroup H := 7 : (S3 3) < He < M. This subgroup has a unique conjugacy class of involutions, which belong× to the class 2B in M. It has six conjugacy classes of elements of order 7. Four of these, namely those labelled in GAP [1, 5] as 7A, 7D, 7E and 7F, belong to the class 7B in M. Since χ(2B) = χ(7B) = 0 and the restriction of χ to H must be a linear combination with non-negative integer coefficients of the irreducible 3-modular Brauer characters of H, we obtain a system of equations for the multiplicities of the characters appearing in this restriction. It suffices to consider the classes 1A, 2A, 7A, 7D and 7F, which yield 5 equations in 10 unknowns. This system has no non-trivial solutions, so we have a contradiction. Therefore, G ∼= M. 4. Proof of Theorem 1.1 — B

Recall that B = 241 313 56 72 11 13 17 19 23 31 47. The prime graph Γ(B) has two isolated vertices| | 31· and· ·47 ·, and· all· other· · vertices· · lie in a connected component. { } { } Suppose that G is a finite group with Γ(G) = Γ(B). Then [4, Theorem 3] implies that G/F (G) = B and that F (G) is a 2-group. If O2(G) = 1, then every element of order ∼ 6 31 in G must act fixed-point freely on O2(G). In particular, every such element in a maximal subgroup L2(31) of G/F (G) ∼= B must act fixed-point freely on O2(G). Let V be a minimal normal subgroup of O2(G), and let χ be the corresponding Brauer character for B. Consider V L2(31). By the 2-modular Brauer character table of L2(31), the ↓ only irreducible modules for L2(31) in characteristic 2 on which elements of order 31 act fixed-point freely are the two modules of dimension 15. There are two conjugacy classes 4 MELISSA LEE AND TOMASZ POPIEL of elements of order 31 in L2(31), both of which belong to the class 31A in B. Each of these classes has Brauer character value c = ( 1+ √ 31)/2 on one of the 15-dimensional modules, and c = ( 1 √ 31)/2 on the other.− In particular,− we infer that − − − ′ ′ χ(31A) = ck + ck for some non-negative integers k and k . (1) 30 Now consider the restriction of V to the maximal subgroup H := [2 ].L5(2). There are 16 irreducible 2-modular Brauer characters for H. Let ai denote the number of composition factors of V H that are isomorphic to the ith irreducible module for H according to the labelling in the↓ 2-modular Brauer character table for H in GAP [1, 5]. The requirement that all elements of order 31 in H act fixed-point freely implies that ai = 0 for all i 1, 6, 11, 12, 13, 14, 15, 16 . The remaining possibilities for the composition factors are the∈ two{ modules of dimension} 5, the two modules of dimension 10, and the four modules of dimension 40. We now construct a system of equations in the remaining multiplicities ai, i I := 2, 3, 4, 5, 7, 8, 9, 10 , as follows. Consider the conjugacy class 31A in H. Elements in∈ this class{ belong to the} class 31A in B. The complex number c in (1) is a sum of 15 distinct primitive 31st roots of unity, and its complex conjugate c is a sum of the other 15 primitive roots. Since the roots are linearly independent over Z, (1) gives us the unique linear combination of the roots equal to χ(31A). This yields a set of 30 linear equations (one per primitive root) with non-negative integer coefficients in the 8 unknowns ai, i I. ∈ This system has no non-trivial solutions, so we have a contradiction. Therefore, G ∼= B. Acknowledgements

This work was initiated in preparation for a research retreat run by the Algebra & Com- binatorics research group of The University of Auckland’s Department of Mathematics, on Waiheke Island in July 2021. We thank all of our fellow participants for a fun and mathe- matically productive retreat, and in particular for discussions with various group members on the recognisability-by-prime-graph problem. Special thanks go to Jeroen Schillewaert for organising the retreat, and to Eamonn O’Brien and Gabriel Verret for helpful discussions about the questions that we have managed to answer here.

References [1] T. Breuer, CTblLib — a GAP package, Version 1.3.2, 2021; http://www.math.rwth-aachen.de/~Thomas.Breuer/ctbllib/ [2] P. J. Cameron and N. V. Maslova, “Criterion of unrecognizability of a finite group by its Gruenberg– Kegel graph”, preprint, 2021; https://arxiv.org/abs/2012.01482v2 [3] J. H. Conway, R. T. Curtis, S. P. Norton, R. A. Parker and R. A. Wilson, ATLAS of finite groups, Oxford University Press, Oxford, 1985. [4] M. Hagie, “The prime graph of a sporadic simple group”, Comm. Algebra 31 (2003) 4405–4424. [5] The GAP Group, GAP — Groups, algorithms, and programming, version 4.11.1, 2021; https://www.gap-system.org [6] A. S. Kondrat’ev, “On the recognizability of sporadic simple groups Ru, HN, F i22, He, McL and Co3 by the Gruenberg–Kegel graph”, Trudy Inst. Mat. i Mekh. UrO RAN 25 (2019) 79–87. (In Russian.) ′ [7] A. S. Kondrat’ev, “On recognition of the sporadic simple groups HS, J3, Suz, O N, Ly, T h, F i23, ′ and F i24 by the Gruenberg–Kegel graph”, Sib. Math. J. 61 (2020) 1087–1092. [8] S. P. Norton and R. A. Wilson, “A correction to the 41-structure of the Monster, a construction of a new maximal subgroup L2(41) and a new Moonshine phenomenon”, J. London Math. Soc. 87 (2013) 943–962. [9] Wolfram Research, Inc., Mathematica, Version 12.3.1, Wolfram Research, Inc., 2021; https://www.wolfram.com/mathematica [10] A. V. Zavarnitsine, “Recognition of finite groups by the prime graph”, Algebra and Logic 45 (2006) 220–231.

Department of Mathematics, University of Auckland, Auckland, New Zealand Email address: [email protected], [email protected]