Local Fusion Graphs and Sporadic Simple Groups

For a group G with G-conjugacy class of involutions X, the local fusion graph F(G,X) has X as its vertex set, with distinct vertices x and y joined by an edge if, and only if, the product xy has odd order. Here we show that, with only three possible exceptions, for all pairs (G,X) with G a sporadic simple group or the automorphism group of a sporadic simple group, F(G,X) has diameter 2.


Introduction
Suppose that G is a finite group with X a G-conjugacy class of involutions (that is, a G-conjugacy class of elements of order 2).The local fusion graph, F(G, X), is the graph whose vertex set is X with distinct vertices x and y joined by an edge whenever xy has odd order.Equivalently, x and y are joined if x, y is a dihedral group of order 2m, m odd, in which case x and y are conjugate in x, y , explaining the graph's epithet.It is clear that G induces graph automorphisms (by conjugation) on F(G, X) and acts transitively on the vertices.Various properties of local fusion graphs have been investigated in [1] and [2].In [2] local fusion graphs for finite symmetric groups are studied, the main result being that they always have diameter two, provided that the degree is at least five.The other finite irreducible Coxeter groups are dealt with in [1], which also considers the possible diameters.There, examples are given of groups which have local fusion graphs whose diameter can be arbitrarily large.Local fusion graphs have even being pressed into service [3] in the area of computational algebra, while graphs of a similar nature appear in [10].
Our main result determines the diameter of the local fusion graphs for (most of) the sporadic simple groups and their automorphism groups.We shall follow the notation and conventions of the Atlas [7] and also use it as a source of data on the sporadic simple groups -Diam(F(G, X)) will denote the diameter of F(G, X).
In Theorem 1 we note that [G : K] = 1 or 2 (see [7]), so G = K or Aut(K).Before outlining the contents of this paper we introduce some more notation.Suppose that G is a finite group and X is a G-conjugacy class of involutions.Let x, y ∈ F(G, X) and i ∈ N ∪ {0}.We shall use d(x, y) to denote the distance between x and y in F(G, X), and the i-th disc of F(G, X), Ξ i (x), is defined by So Ξ 0 (x) = {x}, while Ξ 1 (x) consists of all the neighbours of x in F(G, X).From now on we fix t ∈ X.For a G-conjugacy class C we put and note that X C is invariant under the action of C G (t) by conjugation.We shall sometimes adapt the Atlas [7] notation for conjugacy classes by adding a subscript which indicates the group whose conjugacy class this is.So, for example, 2C HS:2 indicates that we are considering the 2C conjugacy class (as in the Atlas) of HS : 2.
For most of the sporadic groups, calculations employing GAP [18] and Magma [6] yield the diameter of F(G, X) -the details of these being given in Section 2.1.When (G, X) = (B, 2A) or (M, 2A), by extracting appropriate subgroup information from the Atlas we demonstrate in Propositions 4 and 6 that F(G, X) has diameter 2. This approach works largely for the following disparate reasons: the number of C G (t)-orbits of X is small; these orbits are of the form X C for some conjugacy class C of G; and it is possible to identify G-conjugacy classes in smaller subgroups.However, in the case of (G, X) = (B, 2B), for example, X 2B is not a C G (t)-orbit, so we choose in Proposition 5 to investigate F(G, X) using the detailed description of the point-line collinearity graph given in [14].As a by-product, for this case our proof is computer-free.Furthermore, this angle of attack will undoubtedly lead to a sharper picture of the local fusion graph for (M, 2B).Indeed, for (G, X) = (M, 2B) it can be shown that the diameter of the local fusion graph is at most 6.This follows from [13], where it is shown that the commuting involution graph of M on the 2B conjugacy class has diameter 3, when combined with the observation that two commuting 2B involutions are distance 2 apart in F(M, 2B).However, this bound is almost certainly not the best possible.Finally, we remark that for the three graphs not covered by Theorem 1 the permutation rank of G on X is very large.
We thank the referee for their careful reading of this paper, and their many helpful suggestions.

Diameter of F (G, X)
Quite a number of the sporadic simple groups and their conjugacy classes will be dealt with using the next two results.The first is a consequence of some well-known character theoretic results and the second is an elementary observation relating to the size of the first disc of a regular graph.For Lemma 2 we require some more notation, so suppose G is a finite group, with conjugacy classes K 1 , . . ., K and corresponding class sums K 1 , . . ., K in the group algebra CG.Also let a ijk be defined by The a ijk are referred to as the structure constants of G, and may be calculated from the character table of G (see Chapter 28 of [11] for further details).
Lemma 2. Suppose G is a finite group with X a G-conjugacy class of involutions.Assume that X = K i .Then for x ∈ X we have where the sum is over all j such that the conjugacy class K j contains elements of odd order (excluding the conjugacy class of the identity element).
Proof.Let x ∈ X.Then a jii is the number of pairs (z, y) where z ∈ K j and y ∈ K i = X are such that zy = x.So, letting K j run over all G-conjugacy classes of non-trivial odd order elements, j a jii is the number of y ∈ X such that xy has odd order, whence the lemma holds.Lemma 3. Suppose that X is a finite regular graph with valency d.If d > |X |/2, then X is connected and has diameter at most 2.
the regularity of X implies connectedness.Suppose there exists y ∈ X such that x and y are distance 3 apart.Then ∆(x) ∩ ∆(y) = ∅.Therefore Note that Lemma 3 is best possible, as the example of dumbbell graphs attest.

K is not isomorphic to B or M
In Table 1 we list the first disc sizes for the local fusion graphs of G where G = K or Aut(K), K a sporadic simple group.These values have been calculated using Lemma 2 with the aid of GAP [18].Note that for , and so by Lemma 3 we have Diam(F(G, X)) = 2. Now suppose that (K, X) does not fall into this category, and also that K = B or M.
Here our strategy is as follows.First we obtain (by means detailed below) a set of C G (t)-orbit representatives for X.Then for each C G (t)-orbit representative x for which tx has even order, we check that there exists another C G (t)-orbit representative y for which both ty and yx have odd order, thus demonstrating that d(t, x) = 2. Then by the vertex-transitivity of F(G, X) we have Diam(F(G, X)) = 2.For a number of cases it is straightforward to obtain a set of C G (t)-orbit representatives using the Magma [6] command DoubleCosetRepresentatives to find representatives of the double cosets C G (t)gC G (t), where g ∈ G. However this command can fail when the index [G : C G (t)] becomes relatively large.Therefore, when (K, X) = (F i 22 , 2C), respectively (F i 23 , 2B), (F i 23 , 2C), (F i 24 , 2A), (F i 24 , 2B) and (F i 24 , 2D), we use the C G (t)-orbit representatives calculated on page 119, respectively pages 128, 129, 140, 82 and 83 of [17] to verify Theorem 1, while for K = Co 1 we make use of the representatives calculated in [4].Finally, when K = HN and T h the required representatives are taken from [16].
Proof.Here we have G = K ∼ = B.By page 216 of [7], C G (t) has five orbits on X, and they are {t}, X 2B , X 2C , X 3A and Considering elements of order 22 and using [7] we infer that L∩2A = ∅ = L∩2B.Since there are involutions in L of cycle type 2 2 whose product has cycle type 2 2 and X 2A = ∅, we must have Without loss we may take t = (1, 2) ∈ L and then for x = (3, 4) Looking at elements of order 26 and employing [7] again we see that Again, taking t = (1, 2) ∈ L and x = (3, 4) ∈ L we get x ∈ X 2C and then arguing as in the case of X 2B gives X 2C ⊆ Ξ 2 (t).
From [7], looking at elements of order 10 we see X ∩ C G (ξ) = ∅ for ξ ∈ 5A.Now, by [7] C G (ξ) = ξ × L where L ∼ = HS : 2. So we may suppose that t ∈ L. Looking at products of conjugate involutions in HS : 2 we see that t ∈ 2C HS:2 and there exists x ∈ X 4B ∩ L (see, for example [5], Table 2).Employing Magma [6], and using the 100 degree permutation representation of HS : 2, we check that d(t, x) = 2 (note that t has 30 fixed points in this permutation representation), which completes the proof.
We shall use G to denote the point-line collinearity graph of Γ, the maximal 2-local geometry, for G ∼ = B.The data arrayed in [14] and [15] which describes the structure of the graph G will form the backbone of the proof of Proposition 5, and we recommend that the reader has these sources to hand as they are referenced extensively.The vertex set of G is X = 2B.For x ∈ X, the edges of G joined to x are encoded by the lines in Γ x , the residue geometry at x. Now the lines in Γ x correspond to certain type-2 vectors in the Leech lattice (see [14] and [15] again).We shall display these vectors by writing their co-ordinates on a 24-element set which we denote by Ω x (the subscript x is to indicate that we are working in Γ x , as we will be considering a number of different vertices of G).Blank entries mean the co-ordinate is zero.Further, the Steiner system S(24, 8, 5) on Ω x plays an important role and, just as in [14] and [15], we employ Curtis's MOG (see [9] and also [8]) to describe this Steiner system.Proposition 5. Suppose that K ∼ = B and X = 2B.Then Diam(F(G, X)) = 2.
There are three C G (t)-orbits remaining which require our attention, namely ∆ 1 3 (t), ∆ 2 3 (t) and ∆ 3 3 (t).In dealing with these we shall first prove the following.
the electronic journal of combinatorics 22(3) (2015), #P3.18 By Theorem 10 of [14] we have that C G (t)∩C G (x 2 )∩C G (x 3 ) ∼ M 22 : 2 and x 3 +x 2 ∈ HS x 3 (the subscript x 3 telling us that this set of lines are to be viewed in Γ x 3 , the residue of x 3 ).Using the explicit description of HS given in (3.8) of [15] we may, without loss, assume that x 3 + x 2 = 4v ∞ + 4v 14 .By hypothesis x 3 ∈ ∆ 3 2 (x 1 ), and so, relative to x 1 , the C G (x 1 ) ∩ C G (x 3 ) orbits of lines incident with x 3 are listed in Theorem 5 of [14].The description of such C G (x 1 ) ∩ C G (x 3 ) orbits revolves around a certain element of Ω x 3 \ {∞, 14}.Since C G (t) ∩ C G (x 2 ) ∩ C G (x 3 ) acts transitively on Ω x 3 \ {∞, 14}, we may suppose this element is 0 (and replace x 1 by x g 1 , for some g ∈ C G (t) ∩ C G (x 2 ) ∩ C G (x 3 )).Consulting Theorem 5 of [14] again (applied with x g 1 = t and x 3 = x) we see that the lines in (α 3 , x 3 + x 2 , ±3) have one point in ∆ 3  2 (x g 1 ) and the other two are in ∆ 4 3 (x g 1 ).Let ∈ Γ 1 (x 3 ) correspond to the following type-2 vector , and let x 3 , x 4 , x 5 be the three points collinear with .Then, as ∈ (α 3 , x 3 + x 2 , ±3), we have x 4 , x 5 ∈ ∆ 4 3 (x g 1 ).We now wish to determine the C G (t)-orbits to which x 4 and x 5 belong.This can be done by pinning down which C G (t) ∩ C G (x 3 ) orbit belongs to and applying Theorem 10 of [14] )-orbits on lines at x 3 we see the only possibility is that ∈ [0, 8, 28, 64] HSx 3 .Hence by Theorem 10 of [14], one of x 4 and x 5 is in ∆ 2 3 (t) and the other is in ∆ 4 (t), which proves (5.4).
∆ 1 3 (t) ∪ ∆ 3 3 (t) ⊆ Ξ 2 (t). (5.5) the electronic journal of combinatorics 22(3) (2015), #P3.18 [7] the eleventh power of any element of G of order 44 is in 4A.Also from [7], G has only one conjugacy class of elements of order 11.Let g be an element of G of order 11.Then C G (g) = g × M with M ∼ = M 12 , again by [7].So M ∩ 4A = ∅.Moreover, looking at elements of order 22 and using [7] once more we deduce that M ∩ 2A = ∅ = M ∩ 2B.Since elements of order 4 in M 12 square to the class 2B M 12 (the 2B class in M 12 ) and, in G, 4A elements square into 2B, we see that 2A ∩ M = 2A M 12 and 2B ∩ M = 2B M 12 .Hence, if x ∈ X 2A ∪ X 2B ∪ X 4A , we may without loss suppose that t, x M (see, for example, Table 2, line 2 of [5]).Then, by Section 2.1, d(t, x) = 2. Now suppose that x ∈ X 4B .Consulting page 234 of [7] we see that G contains a subgroup H where H ∼ = A 6 , the involutions of H are in 2A and the order four elements of H are in 4B.Thus, without loss of generality, t, x H, whence d(t, x) = 2 by Theorem 1.1 of [2].
Finally we assume that x ∈ X 6A .Put z = tx and H = N G ( z 2 ).By [7], z 2 ∈ 3A and hence H ∼ 3 .F i 24 .Set H = H/ z 2 .In H \ H there are two H involution conjugacy classes, namely 2C F i 24 and 2D F i 24 .Looking at the possible product orders of involutions we see that X ∩ H = 2C F i 24 (the Fischer transpositions).So t and x are transpositions in F i 24 .Thus we may find y ∈ X ∩ H for which ty and yx both have order 3. Consequently ty and yx have odd order (in fact order 3) and so d(t, x) = 2, whence Proposition 6 holds.

Discs of F (G, X)
The disc sizes for the local fusion graphs featuring in Theorem 1 are given in Table 1.
. Note that / ∈ HS x 3 .The inner product of v with the following six type-2 vectors in HS x 3 Thus for at least six of the lines k in HS