The maximal subgroups of the Baby Monster, I

Robert A. Wilson

School of Mathematics and Statistics, The University of Birmingham, Edgbaston, Birmingham B15 2TT

Accepted for publication in the Journal of Algebra, as of 8th June 1998

Abstract

In this paper we describe the completion of the determination of the maximal subgroups of the Baby Monster simple group. Our results are proved using computer calculations with matrix generators for the group. The full details of the calculations will appear elsewhere.

1 Introduction

This paper is a sequel to [8], [10], [12], in which a partial classification of the maximal subgroups of the Baby Monster simple group B of Fischer (see [2]) was achieved. The cases left open after [12] were the 2-local subgroups, and the nonlocal subgroups whose socle is a simple group, isomorphic to one of the following 16 groups.

A_{6}, L_{2}(11), L_{2}(16), L_{2}(17), L_{2}(19), L_{2}(23), L_{2}(25), L_{3}(3), U_{3}(3),
L_{3}(4), U_{3}(8), U_{4}(2),^{2}F_{4}(2)′, G_{2}(3), M_{11} and M_{22}.

The maximal 2-local subgroups have recently been completely classified by Meierfrankenfeld and Shpektorov, with a very beautiful argument [4]. In this paper we deal with the remaining non-local subgroups, using computer calculations with the matrix generators for B constructed in [9]. Here we give the theoretical parts of the argument, together with brief descriptions of the computations and results. We make much use of the information in the Atlas [1], including the character table computed by David Hunt [3]. Full details of the calculations will appear elsewhere [13].

Acknowledgements. The calculations described in this paper were begun on a SUN SPARCstation ELC, provided by the School of Mathematics and Statistics of the University of Birmingham with assistance from a grant from the Science and Engineering Research Council Computational Science Initiative. They were completed on a DEC Alpha AXP workstation provided from another SERC CSI grant. I am grateful to both institutions for their financial assistance.

2 Statement of results

We prove the following theorem.

Theorem 2.1 The maximal subgroups of B are the conjugates of

1. the 22 maximal subgroups listed in the ATLAS;

2. the following 8 new maximal subgroups:

(a) (S_{6} × L_{3}(4):2).2

(b) (S_{6} × S_{6}):4

(c) L_{2}(31)

(d) L_{2}(49).2

(e) L_{2}(11):2

(f) M_{11}

(g) L_{2}(17):2

(h) L_{3}(3)

This may be re-stated as follows.

Theorem 2.2 There are just 30 conjugacy classes of maximal subgroups of B, as listed in Table 1.

3 General strategy

We work our way up from small subgroups to large ones. Suppose we wish to classify up to conjugacy all subgroups of B isomorphic to a particular simple group S. Then we choose some way of generating S by subgroups H and K with L ≤ H ∩ K. Usually we take L = H ∩ K, and often L is a normal subgroup of K, but these are computational conveniences and not always necessary. Our aim is then to classify up to conjugacy all occurrences of such an amalgam (H, K)_{L} in B, and identify which of these generate subgroups isomorphic to S.

Table 1: The maximal subgroups of the Baby Monster

Group  Order
2.^{2}E_{6}(2):2  306 129 918 735 099 415 756 800
2^{1+22}.Co_{2}  354 883 595 661 213 696 000
F i_{23}  4 089 470 473 293 004 800
2^{9+16}S_{8}(2)  1 589 728 887 019 929 600
T h  90 745 943 887 872 000
(2^{2} × F_{4}(2)):2  26 489 012 826 931 200
2^{2+10+20}(M_{22}:2 × S_{3})  22 858 846 741 463 040
2^{5+5+10+10}L_{5}(2)  10 736 731 045 232 640
S_{3} × F i_{22}:2  774 741 019 852 800
2^{[35]}(S_{5} × L_{3}(2))  692 692 325 498 880
HN:2  546 061 824 000 000
O_{8}^{+}(3):S_{4}  118 852 315 545 600
3^{1+8}:2^{1+6}.U_{4}(2).2  130 606 940 160
(3^{2}:D_{8} × U_{4}(3).2^{2}).2  1 881 169 920
5:4 × HS:2  1 774 080 000
S_{4} ×^{2}F_{4}(2)  862 617 600
3^{2+3+6}(S_{4} × 2S_{4})  204 073 344
S_{5} × M_{22}:2  106 444 800
(S_{6} × L_{3}(4):2).2  58 060 800
5^{3}.L_{3}(5)  46 500 000
5^{1+4}:2^{1+4}A_{5}.4  24 000 000
(S_{6} × S_{6}).4  2 073 600
5^{2}:4S_{4} × S_{5}  288 000
L_{2}(49).2  117 600
L_{2}(31)  14 880
M_{11}  7 920
L_{3}(3)  5 616
L_{2}(17):2  4 896
L_{2}(11):2  1 320
47:23  1 081

Typically our strategy consists of finding first H (inside some known proper subgroup), then L (inside H—this is usually easy), and then N_{B}(L) (this is much harder), and finally searching through all possibilities for K inside N_{B}(L).

We describe first the cases where we build up from an A_{5}: in section 4 we consider the four ‘small’ cases A_{6}, L_{2}(11), L_{2}(19) and L_{3}(4), and in section 5 we consider the six ‘large’ cases L_{2}(16), L_{2}(25), U_{4}(2), ^{2}F_{4}(2), M_{11} and M_{22}. Then we consider the progressively more difficult cases L_{2}(23), G_{2}(3), U_{3}(8), L_{3}(3), U_{3}(3) and L_{2}(17).

4 Groups generated by an (A_{5}, A_{5}) amalgam

Many of the groups in the above list can be generated by two subgroups A_{5} intersecting in D_{10}. This includes the groups A_{6}, L_{2}(11), L_{2}(19), and L_{3}(4). It also includes L_{2}(31) and L_{2}(49), and thus our present work provides another proof of the results in [10]. In all cases, by the results of [12], we are only interested in groups containing 5B-elements. We show the following.

Proposition 4.1 If H is a subgroup of B containing 5B-elements, and H is isomorphic to A_{6}, L_{2}(11), L_{2}(19) or L_{3}(4), then H is conjugate to one of the following five groups.

1. H_{1} ≅ A_{6}, with N(H) ≅ M_{10} < T h,

2. H_{2} ≅ A_{6}, with N(H) ≅ A_{6}.2^{2} < (S_{6} × S_{6}).4,

3. H_{3} ≅ A_{6}, with N(H) ≅ M_{10} contained in a 2-local subgroup,

4. H_{4} ≅ L_{2}(11), with N(H) ≅ L_{2}(11):2 maximal,

5. H_{5} ≅ L_{2}(19), with N(H) ≅ L_{2}(19):2 < T h.

Now there are two classes of A_{5} containing 5B-elements, and any such A_{5} has normalizer S_{5}. They can be distinguished in various ways. One type of S_{5} is contained in the Thompson group, and therefore all its involutions are of class 2D in B. The other type has transpositions in class 2C (this fact was noted by Norton [5], and we have also verified it computationally), and the A_{5} can be found in L_{2}(31) for example (although we cannot prove this until the end of the present section!). We therefore have three cases to consider, according as the two groups A_{5} are both of one type, both of the other type, or one of each type.

The case when both A_{5}s are of T h-type. Our plan is to find such an A_{5}, and then find the normalizer in B of a D_{10} in the A_{5}. This group has the shape (D_{10} × 5:4 × 5:4).2. The possibilities for the second A_{5} are then exactly the 400 conjugates of the first A_{5} by elements of the D_{10}-centralizer 5:4 × 5:4.

The results of this analysis are that we obtain one class each of A_{6}, L_{2}(11), L_{2}(19), and L_{2}(49). Each has index 2 in its normalizer. The groups A_{6}.2 ≅ M_{10} and L_{2}(19):2 that we obtain are both contained in T h, while the groups L_{2}(11):2 and L_{2}(49).2 will both turn out to be maximal.

Finding the other type of A_{5}. We have observed that the other type of A_{5} is contained in L_{2}(31). For later convenience we want this to have a D_{10} in common with the first A_{5}. We therefore seek first an L_{2}(31) containing the particular 5B-element which we used earlier. To begin with, we find 31:3 in T h, normalized by this 5-element. Then we look in the normalizer of the 15-element to find an involution extending to L_{2}(31). Finally we conjugate the 31-element by a suitable element to ensure that L_{2}(31) has a D_{10} (not just a 5) in common with the original A_{5}.

The remaining cases. When both A_{5}s are of the second type, we find that such an A_{5} extends to just three groups A_{6}, and not to any of the other groups on the list.

One of these groups extends to S_{6}, and therefore has normalizer A_{6}.2^{2} in the Baby Monster. This is contained as a diagonal subgroup in the maximal subgroup (S_{6} × S_{6}).4. The other two groups are interchanged by a known element, and each has normalizer A_{6}.2 ≅ M_{10}. It can easily be shown that there is a proper subgroup (actually contained in a 2-local subgroup) containing both of these groups M_{10}. In particular, such an A_{6}-normalizer is not maximal.

In the mixed case, we find nothing except the single class of maximal subgroups L_{2}(31).

5 Other groups containing A_{5}

The groups L_{2}(16), L_{2}(25), U_{4}(2), ^{2}F_{4}(2)′, M_{11} and M_{22} all contain A_{5} and can now be classified fairly easily given the work we have already done. The case U_{4}(2) is easy, since U_{4}(2) contains an S_{6}, which in turn contains an S_{5} all of whose involutions are in the same U_{4}(2)-class. It was noted by Norton (see [12]) that the only 5B-type S_{5} with all involutions conjugate in B is the one in T h. But we have already seen that this A_{5} extends to exactly two (conjugate) groups A_{6}, each with normalizer M_{10} (contained in T h). In particular, the S_{5} does not extend to S_{6}, so a fortiori does not extend to U_{4}(2).

Similarly, the Tits group ^{2}F_{4}(2)′ contains an S_{6} all of whose involutions are conjugate, so for the same reason there is no 5B-type^{2}F_{4}(2)′ in B. (Note that this will also follow from the non-existence of a 5B-type L_{2}(25), which is demonstrated below.) There is no 5B-type M_{22} in B since there is no 5B-type L_{3}(4), whereas M_{22} contains a subgroup L_{3}(4).

The remaining cases were dealt with by more computations. First, to classify subgroups M_{11}, take L_{2}(11) and extend A_{5} to S_{5}. There is a unique such group, which can easily be verified to be M_{11}.

Subgroups isomorphic to L_{2}(16). The group L_{2}(16) may be generated from A_{5} by extending D_{10} to D_{30}. Since the centralizer of the D_{10} has order 400, while the centralizer of the D_{30} has order 2, there are just 200 possibilities for the extending element of order 3, falling into 100 inverse pairs. If the A_{5} we start with is the subgroup of T h produced above, then it turns out that none of the 100 extensions of this type is L_{2}(16).

If we start with the other A_{5}, then all but four of the 100 extensions are easily shown not to be L_{2}(16). Now extending the A_{5} to S_{5} cannot normalize any L_{2}(16), since L_{2}(16):2 does not contain S_{5}. Therefore every 5B-type L_{2}(16) in B is self-normalizing, and occurs twice in our list. In particular, there are at most two classes of such subgroups L_{2}(16) in B.

To each of these groups we adjoin the unique involution commuting with the D_{30}. In each case we obtain a group which is larger than L_{2}(16) (since it contains elements of order 34), but is still a proper subgroup of B (since the representation is reducible for this subgroup). It follows that no L_{2}(16) is maximal in B. It is not too hard to tighten up this result to the following.

Proposition 5.1 There are exactly two classes of 5B-type L_{2}(16)s in B, each self-normalizing and contained in the maximal subgroup 2^{9}.2^{16}.S_{8}(2).

It should also be noted that L_{2}(31) can be generated in the same manner, and turns up four times in the list. In particular, we see that L_{2}(31) contains representatives of both classes of 5B-type A_{5}s, from which it immediately follows that it is self-normalizing.

Subgroups isomorphic to L_{2}(25). The group L_{2}(25) may be generated from A_{5} by extending D_{10} to 5^{2}:2. Indeed, L_{2}(25) contains S_{5}, all of whose involutions are conjugate in L_{2}(25), so such an A_{5} must be contained in a copy of T h. Moreover, the 5^{2}-subgroup must be 5B-pure, which implies from [8] that it is contained in the normal 5^{1+4} of the centralizer of any of its non-trivial elements.

In fact there are just four extensions of this type, none of which gives rise to L_{2}(25). Since the Tits group ^{2}F_{4}(2)′ contains L_{2}(25), we obtain another proof that there is no subgroup ^{2}F_{4}(2)′ containing 5B-elements in B.

6 Subgroups isomorphic to L_{2}(23)

Any subgroup L_{2}(23) can be generated by a Frobenius group 23:11 together with an involution inverting any particular element of order 11 in 23:11. We therefore proceed by first finding the 23-normalizer 2 × 23:11 inside the 2B-centralizer 2^{1+22}.Co_{2}. Then we choose an element of order 11 in this group and find the group D_{22} × S_{5} of all elements which centralize or invert it. Finally, we run through the 1 + 10 + 15 = 26 ways of extending 11 to D_{22} to see which extend 23:11 to L_{2}(23). We obtain the following.

Proposition 6.1 There is exactly one class of subgroups L_{2}(23) in B. Any such subgroup is self-normalizing, and is contained in a maximal subgroup F i_{23}.

7 Subgroups isomorphic to G_{2}(3)

Our strategy for classifying subgroups isomorphic to G_{2}(3) is to take representatives of the two conjugacy classes of subgroups L_{2}(13), and in each case extend D_{14} to 7:6. Both types of L_{2}(13) can be found inside the maximal subgroup N(3A) ≅ S_{3} × F i_{22}:2.

Now we need to find the centralizers of our two groups D_{14}, working inside the 7-normalizer (7:3 × 2.L_{3}(4).2).2. The first of these centralizers turns out to be 2 × A_{5}, corresponding to an involution in class 2D in L_{3}(4):2^{2}. The other case, however, gives an involution in L_{3}(4):2^{2}-class 2B, whose centralizer in L_{3}(4) is 3^{2}:Q_{8}. Since the pre-image of the latter group contains the 3-element which centralizes the L_{2}(13), it follows that any extension of the second D_{14} to 7:6 necessarily centralizes a 3A-element, and therefore any G_{2}(3) containing the second L_{2}(13) is contained in F i_{22}.

Finally, then, we need to consider the 21 ways of extending the first D_{14} to 7:6. It turns out that only one of these extends L_{2}(13) to G_{2}(3). The others are easily eliminated by finding an element in the group whose order is not the order of any element of G_{2}(3). Since this case centralizes S_{3}, we have proved the following result.

Proposition 7.1 There are exactly two classes of subgroups G_{2}(3) in B. The normalizers are S_{3} × G_{2}(3) and S_{3} × G_{2}(3):2, both of which are contained in S_{3} × F i_{22}:2.

8 Subgroups isomorphic to U_{3}(8)

Our strategy here is to take a subgroup 3×L_{2}(8) and adjoin an involution inverting a diagonal element of order 9. Now the subgroups L_{2}(8) in B were classified in [11], from which it is easy to see that there are just three classes of subgroups 3 × L_{2}(8) in B, with normalizers as follows.

1. L_{2}(8):3 × 2^{2} × S_{3},

2. L_{2}(8):3 × S_{3},

3. L_{2}(8) × S_{3}.

Any subgroup 3 × L_{2}(8) is contained in S_{3} × F i_{22}:2, and the second and third types can be found in the subgroup S_{3} × 2^{6}S_{6}(2) thereof.

Note that this implies that there are exactly three classes of L_{2}(8) in F i_{22}:2, becoming five classes in F i_{22}. More precisely we have the following result.

Proposition 8.1 The Fischer group F i_{22} contains five classes of subgroups L_{2}(8), with normalizers L_{2}(8):3 × 2, L_{2}(8):3 (two classes), and L_{2}(8) (two classes).

In particular, there are two more classes than is claimed in [7]. At first sight, the classification of subgroups G_{2}(3) and S_{6}(2) in [7] depends on the classification of L_{2}(8). However, closer inspection reveals that it really only depends on the classification of subgroups L_{2}(8):3, which were in fact listed correctly in [7]. Thus the lists of subgroups G_{2}(3) and S_{6}(2) of F i_{22} given there are also (I believe) correct.

Returning now to the classification of subgroups U_{3}(8) in the Baby Monster, we first show by explicit calculation that the second and third classes of L_{2}(8) contain elements of F i_{22}-class 9C. (In fact, this can be obtained from the class fusion from 2^{6}:S_{6}(2) into F i_{22}, which is available in the GAP library [6].) It then follows that the corresponding groups 3×L_{2}(8) contain elements of both B-classes 9B (inside L_{2}(8)) and 9A (the diagonal elements). But all cyclic subgroups of order 9 in U_{3}(8) are conjugate, so these groups 3×L_{2}(8) cannot extend to U_{3}(8) in the Baby Monster. We are therefore left with the case when we have an L_{2}(8)×3 of the first type.

This contains just 9B-elements, and the subgroup generated by elements which invert a given 9B-element has the shape D_{18} × 3^{3}:(2 × S_{4}). Checking through the cases we easily eliminate all but two. Both groups are centralized by an involution, and are conjugate to each other, and to the known U_{3}(8) in 2.^{2}E_{6}(2):2. Thus we have the following.

Proposition 8.2 There is a unique conjugacy class of subgroups U_{3}(8) in B. The normalizer of any such subgroup is 2 × U_{3}(8):6 < 2.^{2}E_{6}(2):2.

9 Subgroups isomorphic to L_{3}(3)

First note that the 3-central elements in L_{3}(3) fuse to class 3B in B, since the centralizer is 3^{1+2}. Next, note that the normalizer of an element of order 13 in B is S_{3} × 13:12 < S_{3} × F i_{22}:2. Now by looking in the chain of subgroups

13:3 < L_{2}(13) < G_{2}(3) < O_{7}(3) < F i_{22}

we see that the elements of order 3 in this group 13:3 are of class 3D in G_{2}(3), so class 3G in O_{7}(3) and class 3D in F i_{22}. Then from the character restriction given on p. 200 of [8] we see that both these elements and the diagonal elements of order 3 fuse to class 3B in B. We also have that there are two classes of subgroups 13:3 in B, with normalisers S_{3} × 13:12 and 3 × 13:12 respectively.

In particular, all 3-elements in any subgroup L_{3}(3) are of class 3B, and therefore the subgroups 3^{2}:2S_{4} of L_{3}(3) are contained in one of the three classes of 3B^{2}- normalizer in B. According to [8], these have the following shapes: 3^{2}.3^{3}.3^{6}.(S_{4} × 2S_{4}) (type (a)), (3^{2} × 3^{1+4}).(2^{2} × 2A_{4}).2 (type (b)), and (3^{2} × 3^{1+4}).(2 × 2S_{4}) (type (c)).

We divide the problem up into three cases, as follows. Case 1: the L_{3}(3) contains a 3^{2}:2S_{4} of type (b). Case 2: the L_{3}(3) contains a 3^{2}:2S_{4} of type (c). Case 3: both classes of 3^{2}:2S_{4} in L_{3}(3) are of type (a).

We show first that case 1 cannot happen. Following the notation of [8], we take the 3^{2} to be generated by (0, 1, 1, 1) and (0, i, i, j). Then its centralizer may be generated by (1, 0, 0, 0), (i, 0, 0, 0), (0, 1, −1, 0), (0, i, −i, 0) together with ⟨−, +, +, +⟩ and ⟨+, [figure], +⟩. This is then normalized by a group 2A_{4} generated by ⟨i, −k, −k, −i⟩ and ⟨ω, ω, ω, ω⟩, extending to 2S_{4} by adjoining

\frac{1}{2} ( \begin{matrix}0 & i − j & j − i & 0 \\ j − i & 0 & 0 & i − j \\ i − j & 0 & 0 & i − j \\ 0 & j − i & j − i & 0 \end{matrix} ) .

In particular, we see by direct calculation (by hand!) that there is a unique class of subgroups 2S_{4} in here, in which the central involution is ⟨−, −, −, −⟩, which is in class 2D in B. On the other hand, all the outer involutions are conjugate to the one given above, which commutes with the element

\frac{1}{2} ( \begin{matrix}−1 & 1 & 1 & 1 \\ −1 & −1 & −1 & 1 \\ −1 & 1 & −1 & −1 \\ −1 & −1 & 1 & −1 \end{matrix} )

of order 3, so cannot be in class 2D. (If it were, its product with the central 3-element would be in class 6J, which has centralizer 3^{1+4}.[2^{11}], and in particular centralizes no elements of order 3 outside the 3^{1+8}.) But this now contradicts the fact that all involutions in L_{3}(3) are conjugate.

The other two cases are rather harder. Note first that in case 3, our hypothesis implies that two of the four elementary abelian 3^{2}-groups in the Sylow 3-subgroup 3^{1+2} of L_{3}(3) are contained in the 3^{1+8}, and therefore the whole Sylow 3-group is in 3^{1+8}. In particular, all 3^{2}-subgroups are of type (a).

In both cases 2 and 3, we make heavy use of two maximal 3-local subgroups, namely N(3B) ≅ 3^{1+8}:2^{1+6}.U_{4}(2):2 and N(3B^{2}) ≅ 3^{2}.3^{3}.3^{6}:(S_{4} × 2.S_{4}).

L_{3}(3)s containing a 3^{2}:2S_{4} of type (c). We show that there is a unique class of such subgroups L_{3}(3), and each such subgroup is self-normalizing and maximal in B.

We may consider the 3^{2} to be generated by (0, 1, 1, 1) and (i, 0, 1, −1), centralized by (i, 0, 0, 0), (0, 1, 0, 0), (1, 0, −i, i) and (0, i, i, i), together with ⟨+, −, −, −⟩^{∗}, where ^{∗} denotes the semilinear map 1 ↦ 1, i ↦ −i. The normalizer is then generated by these elements together with ⟨ [figure: i; i], [figure: +; −]⟩, ⟨+, [figure] ⟩ and ⟨−, +, [figure]⟩.

Thus there are two classes of 3^{2}:2S_{4} of this type. One of these has outer involutions conjugate to ⟨−, +, [figure]⟩, which is in class 2C in B, so cannot be in any L_{3}(3). Thus we must have one of the other class of 3^{2}:2S_{4}, which contains a conjugate of ⟨−, −, [figure: −; −]⟩^{∗}, which is in class 2D in B.

Now some straightforward hand calculations show that this 3^{2}:2S_{4} has trivial centralizer in B, as does its subgroup 3^{2}:(2 × S_{3}) ≅ 3^{1+2}:2^{2}. The latter has two normal 3^{2}-subgroups, one of which is the one of type (c) whose normalizer we started with. The other turns out to be of type (a)—it may be possible to prove this by hand, but I used the computer to do this. In particular we have an embedding of our group 3^{2}:(2 × S_{3}) in 3^{2}.3^{3}.3^{6}:(S_{4} × 2S_{4}), the normalizer of a 3^{2}-group of type (a), and we wish to find all groups 3^{2}:2S_{4} which lie between these two groups.

In fact, it turns out that there are just two such groups—it is not too hard to prove this by hand, but we also checked this on the computer. One of these contains 4E-elements, whereas the 3^{2}:2S_{4} of type (c) does not. Since L_{3}(3) contains a single class of elements of order 4, we can eliminate this case. Thus we have a unique group generated by two copies of 3^{2}:2S_{4}, one of type (c), intersecting in 3^{2}:(2 × S_{3}). We made this group, as described in detail below, and found that it was in fact isomorphic to L_{3}(3). It is clear from the construction that it is self-normalizing. On the other hand, all subgroups L_{3}(3) contained in previously known maximal subgroups have non-trivial centralizers. Hence this group is not contained in any previously known maximal subgroup, and is therefore maximal.

L_{3}(3)s in which all 3^{2}s are of type (a). We show that all such groups L_{3}(3) have non-trivial centralizer.

In this case we adopt a different strategy. We start with a group 13:3, and extend a cyclic 3 subgroup to 3^{2}. As we have seen, there are two classes of subgroups 13:3 in B. Also, each cyclic 3 extends to just 2560 groups 3^{2} of type (a). Thus there are 5120 groups ⟨13:3, 3^{2}⟩ to consider. It turns out that most are easily shown not to be L_{3}(3), and all of the few that remain are centralized by either 2 or S_{3}.

10 Subgroups U_{3}(3)

First we note from Norton’s work on the Monster [5] that any 7A-type U_{3}(3) in M either has centralizer (2^{2} × 3^{2})2S_{4} or has type (2B, 3B, 3B, …). In the former case, the centralizer in B is a non-trivial soluble group, and therefore the normalizer of any such U_{3}(3) in B is contained in a local subgroup. In the latter case, any such U_{3}(3) contains an L_{2}(7) of M-type (2B, 3B, 7A). According to Norton, there is a single class of such L_{2}(7) in M. The corresponding subgroups of B are given in [11]. There are two classes, and the L_{2}(7)-normalizers are L_{2}(7):2 × 2^{2} and L_{2}(7):2 × 2 in the two cases. In both cases the involutions in the L_{2}(7) are of class 2D in B.

The first type of L_{2}(7). In U_{3}(3) there is a maximal subgroup L_{2}(7), which contains two classes of S_{4}, which fuse in U_{3}(3). It follows that U_{3}(3) can be generated by two (conjugate) subgroups L_{2}(7), intersecting in S_{4}. Moreover, since in each case the normalizer of the L_{2}(7) realises the outer automorphism, we may assume that the two copies of L_{2}(7) are conjugate by an element of the centralizer of the S_{4}. Our strategy therefore is to find appropriate groups H ≅ L_{2}(7), and then find the centralizer in B of a subgroup K ≅ S_{4} of H, and investigate the groups ⟨H, H^{x}⟩ as x runs through a transversal for C_{B}(H) in C_{B}(K).

The centralizer of the S_{4} turns out to have order 32, in which we find the subgroup of order 4 which centralizes the original L_{2}(7). Therefore there are eight cases to consider. A small calculation shows that none of these groups is U_{3}(3).

The second type of L_{2}(7). Next we consider subgroups which contain an L_{2}(7) of the second type, which has normalizer 2 × L_{2}(7):2 in B, contained in C(2A).

We adopt a slightly different strategy here. Inside the L_{2}(7) we take an S_{3}, and adjoin an element of class 3B which centralizes the S_{3}. It is not hard to see, by looking in the Monster, that there is a unique class of S_{3} of type (3B, 2D) in B, and that the centralizer of such an S_{3} is a group of the shape 3^{4}:2^{1+4}D_{12}. As a check, we note that this accounts for the full structure constant ξ(2D, 2D, 3B) = 1/2^{7}.3^{5}.

We actually found a subgroup of index 2 in the full centralizer of the S_{3}, which was enough for our purposes. A detailed analysis of this group then enables us to list the 3B-elements it contains. There are 48 such elements (i.e. 24 such cyclic subgroups) in the normal 3^{4}. Outside this subgroup there are 288 cyclic subgroups to consider. We find that none of these cases gives rise to U_{3}(3).

Proposition 10.1 Every U_{3}(3) in B has non-trivial centralizer.

11 Subgroups L_{2}(17)

To classify subgroups isomorphic to L_{2}(17), we take a group 17:8 and then extend the 8 to 8:2 ≅ D_{16}. First note that the normalizer of a cyclic group of order 17 in B is (2^{2} × 17:8).2 ≅ \frac{1}{2}(D_{8} × 17:16). Thus there are three conjugacy classes of groups 17:8 in B. It turns out that one of these contains 8Kelements while the other two contain 8M-elements. To determine the number of ways of extending 8 to D_{16}, we use the structure constants of type (2, 2, 8). The only relevant non-zero structure constants are ξ(2C, 2C, 8K) = 512/98304, ξ(2D, 2D, 8K) = 3072/98304, ξ(2C, 2D, 8M) = ξ(2D, 2C, 8M) = 512/32768 and ξ(2D, 2D, 8M) = 2048/32768. Of course, since all involutions in L_{2}(17) are conjugate, the only dihedral groups we are interested in contain only involutions of class 2D. However, it seemed to be more trouble than it was worth to use this fact in the calculations.

In fact, we verified computationally that the three classes of 17:8 contain elements of class 8K, 8M and 8M respectively, but this can also be proved by character restriction, as follows. We have

(2^{2} × 17:8).2 < (2^{2} × S_{4}(4):2).2 < (2^{2} × F_{4}(2)):2,

in which the 8-element in 17:8 is of class 8AB in S_{4}(4):2. In S_{4}(4):4 these elements have square roots, which by character restriction are in class 16HI in F_{4}(2):2. Thus the elements of order 8 are in class 8K in F_{4}(2). Now the ordinary 4371character of B restricts to 2^{2} × F_{4}(2) as 833 + (1 + 1105) + (1 + 1105) + 1326, where the brackets delimit the eigenspaces of the central 2^{2}. Thus, using character values and power maps, the 8K-elements in F_{4}(2) fuse to class 8K in B, while both diagonal classes of 8-elements fuse to 8M in B.

We split the calculation into three cases, according to which of the three elements of order 8 we are using.

The 8K-case. Here the element of order 8 has normalizer of order 2^{17}.3, and there are 2^{12}.3 cosets which consist of inverting elements, and just 448 of these consist of involutions. We ran through all these 448 ways of extending the 8 to a group D_{16}. It turned out that there were just six groups L_{2}(17) among these cases, two centralized by 2^{2} and four with centralizer 2. The first two are therefore the two classes of L_{2}(17) inside F_{4}(2), which are interchanged by the outer automorphism, and have normalizers 2^{2} × L_{2}(17) < (2^{2} × F_{4}(2)):2. Of the last four, two are centralized by each of the two 2A-elements in the 17centralizer. Now ^{2}E_{6}(2) contains four classes of subgroups L_{2}(17), one of which is centralized by an S_{3} of outer automorphisms, while the other three are permuted in the natural manner by this S_{3}. In the Baby Monster, a group 2 of outer automorphisms is realised, so there is the one class of L_{2}(17) in F_{4}(2):2 that we have already seen, and one other, whose normalizer is 2 × L_{2}(17), contained in 2.^{2}E_{6}(2):2.

The first 8M-case. Using the same method as before, we run through all the 384 inverting cosets which consist of involutions, and find four cases which give rise to L_{2}(17). Each has trivial centralizer, which means they are all conjugate, and have normalizer L_{2}(17):2. We verified explicitly the existence of this subgroup L_{2}(17):2, by checking generators and relations in suitable elements of the group. This subgroup is not contained in any of the previously known maximal subgroups of B. It is therefore maximal.

The second 8M-case. Using the same method as before, we then run through the 384 involutory cosets inverting the second 8M-element, and find that none could extend 17:8 to L_{2}(17). (There are however some cases which give 2×L_{2}(17), in which the L_{2}(17) is of 8K-type.)

To summarise the results of this section, we have

Proposition 11.1 There are exactly three conjugacy classes of subgroups isomorphic to L_{2}(17) in B. Two have type (8K, 9B) and have normalizers 2^{2} × L_{2}(17) and 2 × L_{2}(17). The third has type (8M, 9A) and normalizer L_{2}(17):2.

References

[1] J. H. Conway, R. T. Curtis, S. P. Norton, R. A. Parker and R. A. Wilson, An ATLAS of Finite Groups, Oxford University Press, 1985.

[2] B. Fischer, A note concerning {3,4}-transposition groups, unpublished.

[3] D. C. Hunt, The character table of the Baby Monster, unpublished.

[4] U. Meierfrankenfeld and S. V. Shpektorov, The maximal 2-local subgroups of the Monster and Baby Monster, in preparation.

[5] S. P. Norton, The anatomy of the Monster. I, in The ATLAS 10 years on (ed. R. T. Curtis and R. A. Wilson), LMS Lecture Note Series, Cambridge University Press, 1997. (to appear)

[6] M. Schönert et al., GAP—Groups, Algorithms and Programming, RWTH Aachen, 1995.

[7] R. A. Wilson, On maximal subgroups of the Fischer group F i_{22}, Math. Proc. Cambridge Philos. Soc. 95 (1984), 197–222.

[8] R. A. Wilson, Some subgroups of the Baby Monster, Invent. Math. 89 (1987), 197–218.

[9] R. A. Wilson, A new construction of the Baby Monster, and its applications, Bull. London Math. Soc. 25 (1993), 431–437.

[10] R. A. Wilson, Some new subgroups of the Baby Monster, Bull. London Math. Soc. 25 (1993), 23–28.

[11] R. A. Wilson, The symmetric genus of the Baby Monster, Quart. J. Math. (Oxford) 44 (1993), 513–516.

[12] R. A. Wilson, More on maximal subgroups of the Baby Monster, Arch. Math. 61 (1993), 497–507.

[13] R. A. Wilson, The maximal subgroups of the Baby Monster, II, in preparation.