2-local Subgroups Research Articles

Classification of some 3-subgroups of the finite groups of Lie type E6

We consider the finite exceptional group of Lie type G=E6ε(q) (universal version) with 3|q−ε, where E6+1(q)=E6(q) and E6−1(q)=2E6(q). We classify, up to conjugacy, all maximal-proper 3-local subgroups of G, that is, all 3-local M<G which are maximal with respect to inclusion among all proper subgroups of G which are 3-local. To this end, we also determine, up to conjugacy, all elementary-abelian 3-subgroups containing Z(G), all extraspecial subgroups containing Z(G), and all cyclic groups of order 9 containing Z(G). These classifications are an important first step towards a classification of the 3-radical subgroups of G, which play a crucial role in many open conjectures in modular representation theory.

$$\varvec{{\mathbb {Z}}_3}$$ Z 3 -orbifold construction of the Moonshine vertex operator algebra and some maximal 3-local subgroups of the Monster

In this article, we describe some maximal 3-local subgroups of the Monster simple group using vertex operator algebras (VOA). We first study the holomorphic vertex operator algebra obtained by applying the orbifold construction to the Leech lattice vertex operator algebra and a lift of a fixed-point free isometry of order 3 of the Leech lattice. We also consider some of its special subVOAs and study their stabilizer subgroups using the symmetries of the subVOAs. It turns out that these stabilizer subgroups are 3-local subgroups of its full automorphism group. As one of our main results, we show that its full automorphism group is isomorphic to the Monster simple group by using a 3-local characterization and that the holomorphic VOA is isomorphic to the Moonshine VOA. This approach allows us to obtain relatively explicit descriptions of two maximal 3-local subgroups of the shape \(3^{1+12}.2.{{\mathrm{Suz}}}{:}2\) and \(3^8.\Omega ^-(8,3).2\) in the Monster simple group.

A 3-local characterization of M(24)

We identify M(24) by the structure of the normalizer of a 3-central element. Also as a corollary we identify M(24) by two of its 3-local subgroups.

Maximal 2-local subgroups of [formula omitted

Let G=E7(q) be the finite exceptional group of Lie type (universal version). We classify, up to conjugacy, all maximal-proper 2-local subgroups of G, that is, all 2-local M<G which are maximal with respect to inclusion among all proper subgroups of G which are 2-local. For this purpose, we first determine, up to conjugacy, all elementary-abelian 2-subgroups of G containing the center Z(G). These classifications are an important first step towards a classification of the 2-radical subgroups of G.

Maximal 2-local subgroups of [formula omitted] and [formula omitted

We classify maximal 2-local subgroups of the finite groups F4(q) and E6η(q) when q is odd.

Radical 3-subgroups and essential 3-rank of [formula omitted

We classify maximal 3-local subgroups and radical 3-subgroups of G=F4(q) when q odd, and then determine the essential 3-rank of the Frobenius category FD(G), where D is a Sylow 3-subgroup of G.

The Fischer-Clifford Matrices and Character Table of a Maximal Subgroup of Fi24

The Fischer group [Formula: see text] is the largest 3-transposition sporadic group of order 2510411418381323442585600 = 222.316.52.73.11.13.17.23.29. It is generated by a conjugacy class of 306936 transpositions. Wilson [15] completely determined all the maximal 3-local subgroups of Fi24. In the present paper, we determine the Fischer-Clifford matrices and hence compute the character table of the non-split extension 37· (O7(3):2), which is a maximal 3-local subgroup of the automorphism group Fi24 of index 125168046080 using the technique of Fischer-Clifford matrices. Most of the calculations are carried out using the computer algebra systems GAP and MAGMA.

An identification of the Monster group

We identify the Monster from two of its 3-local subgroups.

Groups saturated by finite simple groups

We describe periodic groups saturated by groups in the set ℭ = {L 2(r), r ⩾ 4; L 3(2 m ), m ⩾ 1; U 3(2 m ), m ⩾ 2; Sz(2 m ), m ⩾ 3}. As a corollary we give a description of periodic groups G saturated by finite simple groups and satisfying one of the following conditions: (a) centralizers of involutions in G are 2-closed; (b) G contains a strongly embedded 2-local subgroup.

On fixed point sets and Lefschetz modules for sporadic simple groups

We consider 2-local geometries and other subgroup complexes for sporadic simple groups. For six groups, the fixed point set of a noncentral involution is shown to be equivariantly homotopy equivalent to a standard geometry for the component of the centralizer. For odd primes, fixed point sets are computed for sporadic groups having an extraspecial Sylow p -subgroup of order p 3 , acting on the complex of those p -radical subgroups containing a p -central element in their centers. Vertices for summands of the associated reduced Lefschetz modules are described.

Decomposition numbers of non-principal blocks of [formula omitted] for characteristic 3

In this paper, the decomposition numbers of the non-principal blocks of the largest Janko group J 4 [Z. Janko, A new finite simple group of order 86,775,571,046,077,562,880 which possesses M 24 and the full covering group of M 22 as subgroups, J. Algebra 42 (1976) 564-596] for characteristic 3 are determined up to some unknown parameters. We are also concerned with the decomposition numbers of maximal 2-local subgroups of J 4 in odd characteristics.

A theorem about coprime action

It is well known that if an elementary abelian p-group P acts on a p ′ -group Q and Q = [ Q , P ] , then Q = 〈 [ C Q ( A ) , P ] | A ⩽ P of index p〉. Does a similar statement hold for C Q ( P ) ? Under further assumptions, the answer is yes. Goldschmidt proves theorems of this flavour in [D.M. Goldschmidt, Weakly embedded 2-local subgroups of finite groups, J. Algebra 21 (1972) 341–351. [1]; D.M. Goldschmidt, Strongly closed 2-subgroups of finite groups, Ann. of Math. (2) 102 (1975) 475–489. [2]] and uses them to construct signalizer functors. For the same reason we prove a result of this type, under the assumption that Q is soluble.

An identification of [formula omitted

We identify Co 1 from two of its 3-local subgroups.

Maximal 2-Local Geometries for the Symmetric Groups#

The set 𝒩max (G, T) consisting of all maximal 2-local subgroups of G = Sym(n) which contain T, a Sylow 2-subgroup of G, is investigated. In addition to determining the structure of the subgroups in 𝒩max (G, T), the simplicial sets of maximal rank are classified.

Parabolic 2-local subgroups in groups of finite Morley rank of even type

Parabolic 2-local subgroups in groups of finite Morley rank of even type

The Fischer-Clifford matrices of a maximal subgroup of 𝐹𝑖′₂₄

The Fischer group F i 24 ′ Fi_{24}^{\prime } is the largest sporadic simple Fischer group of order \[ 1255205709190661721292800 = 2 21 .3 16 .5 2 .7 3 .11 .13 .17 .23 .29 . 1255205709190661721292800 = 2^{21}.3^{16}.5^2.7^3.11.13.17.23.29 \;\;. \] The group F i 24 ′ Fi_{24}^{\prime } is the derived subgroup of the Fischer 3 3 -transposition group F i 24 Fi_{24} discovered by Bernd Fischer. There are five classes of elements of order 3 in F i 24 ′ Fi_{24}^{\prime } as represented in ATLAS by 3 A 3A , 3 B 3B , 3 C 3C , 3 D 3D and 3 E 3E . A subgroup of F i 24 ′ Fi_{24}^{\prime } of order 3 3 is called of type 3 X 3X , where X ∈ { A , B , C , D , E } X \in \{A,B,C,D,E \} , if it is generated by an element in the class 3 X 3X . There are six classes of maximal 3-local subgroups of F i 24 ′ Fi_{24}^{\prime } as determined by Wilson. In this paper we determine the Fischer-Clifford matrices and conjugacy classes of one of these maximal 3-local subgroups G ¯ := N F i 24 ′ ( ⟨ N ⟩ ) ≅ 3 7 ⋅ O 7 ( 3 ) \bar {G} := N_{Fi_{24}^{\prime }}(\langle N\rangle ) \cong 3^7{\cdot }O_7(3) , where N ≅ 3 7 N \cong 3^7 is the natural orthogonal module for G ¯ / N ≅ O 7 ( 3 ) \bar {G}/N \cong O_7(3) with 364 364 subgroups of type 3 B 3B corresponding to the totally isotropic points. The group G ¯ \bar {G} is a nonsplit extension of N N by G ≅ O 7 ( 3 ) G \cong O_7(3) .

An identification theorem for groups of finite Morley rank and even type

The paper contains a construction of a definable BN-pair in a simple group of finite Morley rank and even type with a sufficiently good system of 2-local parabolic subgroups. This provides ‘the final identification theorem’ for simple groups of finite Morley rank and even type.

The Alperin Weight Conjecture and Dade's Conjecture for the Simple Group J4

AbstractThe authors construct faithful permutation representations of maximal 2-local subgroups and classify the radical chains of the Janko simple group J4; hence the Alperin weight conjecture and the Dade reductive conjecture for J4 are verified.

Fusion of 2-elements in groups of finite Morley rank

The Alperin-Goldschmidt Fusion Theorem [1, 5], when combined with pushing up [7], was a useful tool in the classification of the finite simple groups. Similar theorems are needed in the study of simple groups of finite Morley rank, in the even type case (that is, when the Sylow 2-subgroups are of bounded exponent, as in algebraic groups over fields of characteristic 2). In that context a body of results relating to fusion of 2-elements and the structure of 2-local subgroups is needed: pushing up, and the classification of groups with strongly or weakly embedded subgroups, or have strongly closed abelian subgroups (c.f, [2]). Two theorems of Alperin-Goldschmidt type are proved here. Both are needed in applications.The following is an exact analog of the Alperin-Goldschmidt Fusion Theorem for groups of finite Morley rank, in the case of 2-elements:Theorem 1.1. Let G be a group of finite Morley rank, and P a Sylow 2-subgroup of G. If A, B ⊆ P are conjugate in G, then there are subgroups Hi ≤ Pand elementsxi ∈ N(Hi) for 1 ≤ i ≤ n, and an elementy ∈ N(P), such that for all i:1. Hi is a tame intersection of two Sylow 2-subgroups;2. CP(Hi) ≤ Hi;3. N(Hi)/Hiis 2-isolatedand(a) (b) .

On the Finite Simple Groups All of Whose 2-Local Subgroups Are Solvable

On the Finite Simple Groups All of Whose 2-Local Subgroups Are Solvable