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The Odd-Local Subgroups of the Monster

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The Odd-Local Subgroups of the Monster J. Austral. Math. Soc. {Series A) 44 (1988), 1-16 THE ODD-LOCAL SUBGROUPS OF THE MONSTER ROBERT A. WILSON (Received 18 September 1985) Communicated by H. Lausch Abstract We determine all conjugacy classes of maximal /?-local subgroups of the Monster for p # 2. 1980 Mathematics subject classification (Amer. Math. Soc.): 20 D 08. 1. Introduction In this paper we find all the maximal /»-local subgroups of the Fischer-Griess Monster group, for each odd prime p. The results for p > 5 are not new—they were first obtained by S. P. Norton, but his proofs have not been published. The hardest case, however, is the case p = 3, and our main result is THEOREM 3. Any 3-local subgroup of the Monster group M is contained in a conjugate of one of the seven groups N(3A) = 3 • FiM, N(3B)~ 3V"12-2 • Suz:2, N(3C) = S3XTh. Conversely, none of these groups is contained in a conjugate of any other. © 1988 Australian Mathematical Society 0263-6115/88 $A2.00 + 0.00 1 Downloaded from https://www.cambridge.org/core. IP address: 170.106.202.58, on 24 Sep 2021 at 12:06:05, subject to the Cambridge Core terms of use, available at https://www.cambridge.org/core/terms. https://doi.org/10.1017/S1446788700031323 2 Robert A. Wilson [2] The proof is based largely on calculations inside the group 31 + 12 • 2Suz, and therefore depends heavily on the simple notation for this group introduced in Section 3 below. This notation can be restricted to various subgroups of 31 + 12 . 2Suz, and enables one to find the maximal 3-local subgroups of B [10], Th [11], Fi24 and Fi23 [12]. It is really an extension and modification of the notation used in [9] to study the 3-local subgroups of Fi22. Other notation is taken from the ATLAS [3]. The results for p = 5 and 7 are THEOREM 5. Let Y be any non-trivial 5-subgroup of the Monster. Then N(Y) is contained in a conjugate of one of the 6 groups: N(5A) = (Dw X HN) • 2, 2 2 = (5 :4-2 X U3(5)):S3, 4 4 ) = 5 :(3 X 2 • L2(25)):2. Conversely, none of these groups is contained in a conjugate of any other. THEOREM 7. Let Y be any non-trivial 1-subgroup of the Monster. Then N(Y) is contained in a conjugate of one of the 5 groups 2 ^ (7 :(3X2^4)XI3(7)):2, Conversely, none of these groups is contained in a conjugate of any other. Note. This last group is missing from the list given in [3]. The results for p > 7 are given in the final section of the paper. 2. Basic 3-local structure There are three classes of elements of order 3 in the Monster, with normalizers = 3\+l2-2Suz:2, JV(3C) = S3 X Th. Downloaded from https://www.cambridge.org/core. IP address: 170.106.202.58, on 24 Sep 2021 at 12:06:05, subject to the Cambridge Core terms of use, available at https://www.cambridge.org/core/terms. https://doi.org/10.1017/S1446788700031323 [ 3 ] The odd-local subgroups of the Monster 3 Of these three classes, just the 32?-elements are 3-central. In particular, the group 31 + 12 • 2Suz contains a Sylow 3-subgroup of M, and hence contains a representa- tive of every conjugacy class of elementary Abelian subgroups. Throughout the rest of Sections 2-8, Y will denote an elementary Abelian 3-group of order at least 32. PROPOSITION 2.1. // Y contains 3C-elements, then N(Y) is contained in N(Y0) where Yo is a non-trivial elementary Abelian 3-group containing no 3C-elements. PROOF. The subgroup Th contains no elements of Af-class 3C, while all diagonal 3-elements in 3 X Th are of M-class 3C. It follows that Y has a unique subgroup Yo of index 3 containing no 3C-elements, so N(Y) < N(YQ), and the proposition is proved. From now on, therefore, we concern ourselves only with the 3A- and 3B- elements, and we begin by studying their centralizers. Restricting the 196883-character of M to 3Fi'24 we obtain la + 8671a + 57477a + 783aZ» + 64584a£>, where the last four characters are faithful and the first three characters are not. Calculating the character values on the various elements of order 3 we deduce the class fusion: 2 Fi'24-cl&ss M-classes 3 -centralizer in M 2 3A 3A3 3 X O+(3) 1 + 10 3B 3BXA2 (3 x3 ):C/5(2) 38 3C 3A,B2 • 2t/4(3) 5 . 14] 3D 3B3 [2 3 2 3E 3C3 3 X G2(3) PROPOSITION 2.2. // Y contains 3A-elements but no 3B-elements, then N(Y) is contained in PROOF. From Proposition 2.1 and the above table it suffices to show that Fi2A contains no 32-group whose non-trivial elements are all of Fz'24-class 3A. This is proved in [12] by restricting the characters of degrees 783 and 2808. From now on, we can assume that Y is an elementary Abelian group contain- ing 35-elements but no 3C-elements. For the rest of the paper we fix a 3B- element x contained in Y, and let X = 31 + 12 be normal in N(x). Downloaded from https://www.cambridge.org/core. IP address: 170.106.202.58, on 24 Sep 2021 at 12:06:05, subject to the Cambridge Core terms of use, available at https://www.cambridge.org/core/terms. https://doi.org/10.1017/S1446788700031323 4 Robert A. Wilson [4] Our next task is to investigate the structure of the 3.6-centralizer in detail. Now the values of the 196883-character on the elements of classes 2B, 3B and 6B show that it restricts to 31 + 12 • ISuz as 143 e 65520 e {(729 © 729') ®(12 © 78)}, where characters are denoted by their degrees, and 12, 78, 143 are faithful irreducible characters of 6Suz, 3Suz and Suz respectively, 729 and 729' are extensions of the faithful irreducible characters of 31 + 12 to 31 + 12 • ISuz, and 65520 is a faithful monomial representation of 312: ISuz. Again we calculate the character values on the various elements of order 3, and deduce that the fusion from 6Suz to M is given by Suz-c\a&s Af-classes 32-centralizer in M 8 3A 3A2B1 3 • 2U4(3) 5 14 3B 3AXB2 [2 • 3 ] 6 3C 3C3 3 .2^6 3. A notation for 3l + n • 2Suz In this section we introduce a notation for the group 312:25MZ = C(x)/(x). Since the normal 312-group is really a reduction of the complex Leech lattice modulo 6 = v^-3~, it is useful to describe this lattice first. More details can be found in [7], though for present purposes it is best to use the "quaternionic" version given in [6], with slight modifications. It must be borne in mind however that the group 6Suz does not preserve the quaternionic structure. We write vectors of the Leech lattice in the 6 by 4 MOG array in the usual way, and associate the rows to the quaternions 1, i, j and k respectively. By interpreting each column of the MOG as a single quaternion, and dividing by 2, we identify the Leech lattice with a certain set of vectors in a quaternionic 5+12 6-space. There is a "monomial" subgroup 2 : 3A6 of the automorphism group of the lattice which preserves the associated decomposition into 6 quaternionic 1-spaces (real 4-spaces). This may be generated by left and right multiplications by diagonal matrices corresponding to "hexacode" words, together with the hexacode automorphism group 3A 6. This multiplicative hexacode may be gener- ated by the elements (j, k, k, j, k, j), (k, j, j, k, k, j), (k, j, k, j, j, k) and their images under a. Here w denotes the map defined by 1 -» 1, i -* j -* k -* i: this may be realized as conjugation by ^(-1 + i + j + k). Full details are given in [8]. We want generators for 6Suz as a subgroup of the lattice automorphism group 5+6 2Cov so we take the group 2 generated by the right multiplications by hexacode words, and extend by the hexacode automorphism group 3A6, which Downloaded from https://www.cambridge.org/core. IP address: 170.106.202.58, on 24 Sep 2021 at 12:06:05, subject to the Cambridge Core terms of use, available at https://www.cambridge.org/core/terms. https://doi.org/10.1017/S1446788700031323 [5 ] The odd-local subgroups of the Monster 5 may be generated by (u, a, +, •—•—••) and ( * «^>C^). Then we need one extra generator, which can be taken to be right multiplication by the matrix Inooool 1-10 0 0 0 J ~ 0 0 110 0 0 0 1-10 0 0 0 0 0 11 0 0 0 0 1-1 Note that since « always denotes conjugation by w = ^(-1 + i + j + k), this group is not a group of right multiplications by quatemionic matrices. Now let 8 = i + j + k, and consider the endomorphism 6 of L defined by multiplying each quatemionic coordinate of a vector on the left by 6. Then L/QL is an Abelian group of exponent 3 and order 312. Now left multiplication by w acts trivially on L/QL, so conjugation by w is the same as right multiplica- tion by w. Hence the group 6Suz can be re-interpreted as a group of right multiplications on L/QL, where the central element of order 3 acts trivially.
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