New Upper Bounds on the Spreads of the Sporadic Simple Groups by Ben Fairbairn | Papers by Ben

NEW UPPER BOUNDS ON THE SPREADS OF THE SPORADIC SIMPLE GROUPS BEN FAIRBAIRN Abstract. We give improved upper bounds on the exact spreads of many of the larger sporadic simple groups, in some cases improving on the best known upper bound by several orders of magnitude. Keywords: exact spread, sporadic simple group MSC: 20D08, 20C15 1. Introduction Recall that a group is said to be 2-generated if it is generated by just two of its elements. Every finite simple group is 2-generated (see [1]) and many authors have considered the question of how easily a pair of elements generating a simple group may be obtained. One quantity measuring this introduced by Brenner and Wiegold in [6] and motivated by earlier work of Binder [2] is the concept of the spread of a group. Let G be a group. We say that G has spread r if for any set of distinct non-trivial elements X := {x1 , . . . , xr } ⊂ G there exists an element y ∈ G with the property that xi , y = G for every 1 ≤ i ≤ r. We say that this element y is a mate of X and that G has exact spread s(G):=r if G has spread r but not r + 1. The concept of spread is also of interest to computational group theorists since it is useful in the the analysis of the celebrated product replacement algorithm for producing random elements of groups [14]. The concept is also of interest when studying the generating graph of a group [16, Section 4]. The exact spreads of the finite simple groups have been much studied [2, 6, 8, 15]. In particular, bounding the value of the exact spreads of the sporadic simple groups has recently been investigated by several different authors [3, 4, 12, 18] and it is these cases that we focus on here. More specifically we prove the following. Theorem 1. The exact spreads of the sporadic simple groups are bounded by the values given in Tables 1 and 2. Most of the bounds listed in Tables 1 and 2 are not new. The upper bounds given in Table 1 were obtained by Bradley and Holmes in [3] using coverings of a group by sets of proper subgroups and as such 1 Table 1. Bounds on s(G) for the smaller sporadic simple groups. The upper bounds are proved in [3]. The lower bound for M11 is proved in [3, 7, 18]. The lower bound for M23 is proved in [11]. All other lower bounds are proved in [8]. (Note that aside from M11 , M12 and J2 the lower bounds stated in [3, Table 1] are incorrect.) G M11 M12 J1 M22 J2 M23 HS Upper bound G Lower bound 3 J3 3 9 M24 3 179 McL 76 26 He 20 24 Suz 5 8064 Co3 8064 33 Fi22 18 Upper bound Lower bound 597 76 56 11 308 70 1223 198 956 40 1839 98 186 13 are unable to handle the sporadic simple groups with very large coverings, which is essentially the larger groups. Our methods are unable to improve upon these bounds. What is new here are the upper bounds listed in Table 2 for all the groups that the methods of [3] could not deal with. The bounds given in Tables 1 and 2 are, as far as the author is aware, the best known, accepting that Bradley and Holmes claim that for the groups they considered “better results were obtained for some of the groups in trial runs, but our table gives only the results that were given by known seeds” [3, p.138]. In Section 2 we will introduce some preliminary ideas that we will use to prove our bounds in Section 3 in every case aside from the Baby Monster and the Monster group that we shall deal with separately in Section 4. 2. Preliminaries In this section we shall describe some concepts that will be useful in proving Theorem 1. 2 Table 2. The best previous upper bounds for the larger sporadic groups proved in [4]; the new upper bounds proved here and the lower bounds proved in [8]. Note that the seemingly better lower bound given in [12] for HN (10999) is incorrect - see [8, Section 4.7]. Old upper bound G Old upper bound New upper bound New upper bound Lower bound Lower bound Ru 12990752 Th 103613642531 1252799 976841774 2880 133997 O’N 5960127 Fi23 8853365473 2857238 31670 3072 911 Co2 5240865 Co1 58021747714 1024649 46621574 270 3671 HN 229665984 J4 251012689269463297 74064374 47766599363 8593 1647124116 Ly 112845651178977 Fi24 309163967798745777216 1296826874 7819305288794 35049375 269631216855 B 3843675651630431666542962843030 3843461129719173164840195954999 174702778623598780219391999999 M 14587804270839626161268024115186834207944682668030 5791748068511982636944259374 3385007637938037777290624 G 2.1. Support. Let G be a group and let x ∈ G# where G# := G \ {e}. We define the support of x to be the set supp(x) := H<G,x∈H H #. In other words y ∈ supp(x) means that y is an element of G# that lies in a proper subgroup containing x and so y cannot be a mate for the set {x}. We extend this to subsets X ⊂ G# as follows: supp(X) := x∈X supp(x). If Y ⊂ supp(X) we say that X supports Y . In particular, elements of supp(X) cannot be a mate to X. 3 Table 3. Support classes for the sporadic simple groups. G Class G Class G Class G Class M11 2A J3 2A O’N 2A Th 2A M12 2A M24 2A Co3 2A Fi23 2A J1 2A McL 2A Co2 2B Co1 2A M22 2A He 2A Fi22 2A J4 2B J2 2A Ru 2B HN 2B Fi24 2B HS 2A Suz 3A Ly 2A M 2B 2.2. Support classes and characters. Continuing the earlier nomenclature, we say a conjugacy class C ⊂ G is a support class if the set C supports the set G# . For each of the bounds that we improve upon here, our improved bound is obtained by showing that some small conjugacy class of G is a support class. Note that not every simple group has a support class e.g. the Baby Monster has no support class as the only maximal subgroups with elements of order 47 are copies of the Frobenius group 47:23, but no proper subgroup containing elements of order 23 contains elements of order 31 - see the list of maximal subgroups given in [17] (the list given in [10] is incomplete). Whilst we cannot improve upon the best known bound in this case precisely the same methods as the other cases, our approach is not entirely redundant here - see Section 4. To obtain a set of elements that has no mate, and thus provide an upper bound on the spread, it is sufficient to take one generating element from each cyclic subgroup generated by an element of a support class. We thus have the following easy lemma. Lemma 2. Let C ⊂ G# be a support class, d := | g ∩ C| for g ∈ C. Then s(G) + 1 ≤ |C|/d. Given a conjugacy class C ⊂ G# we define its support character χC to be the sum of the primitive permutation characters of G that are nonzero on C. Since the transitive permutation character 1 ↑G is nonzero H at a class C if and only if C ∩ H = ∅ we have the following lemma. Lemma 3. A conjugacy class C ⊂ G# is a support class if and only if χC (g) > 0 for every g ∈ G. 3. Computing the bounds Our new upper bounds are obtained by finding a small support class using Lemma 3 and then using this to obtain a bound using Lemma 2. This is easily done using the GAP algebra system [13]; first by obtaining the primitive permutation characters using standard GAP functions (primarily the GAP character table library and the tables of marks), 4 then obtaining the support characters (if any) using this data and finally by reading off the support class that gives the best bound from the list just obtained. The support classes giving the best bound found in this way are given in Table 3. For completeness we give these support classes for each of the sporadic groups that possesses one, that is every sporadic simple group aside from the Mathieu group M23 and the Baby Monster B. For example, in several cases (M11 , J1 , M22 , M23 , J3 , McL, O’N and Ly) there is only one class of involutions and every maximal subgroup has even order (see [10]) so the support character χ2A is the sum of every primitive permutation character and is therefore positive on every class. The class 2A is therefore a support class in these cases by Lemma 3. Note that whilst the Thompson group, Th, also has only one class of involutions it also has a class of maximal subgroups of odd order isomorphic to the Frobenius group 31:15. The elements of this subgroup can easily be seen to also belong to other maximal subgroups with structure 25 .L5 (2) (see [10, p.70]). The support character χ2A is thus equal to the sum of every primitive permutation character, aside from the one defined by the maximal copies of 31:15, and χ2A (g) > 0 for every g ∈ Th. Thus 2A is a support class by Lemma 3. 4. The Baby Monster and the Monster 4.1. The Baby Monster. As noted in Section 2.2 the Baby Monster group B does not have a support class. However, we can use a union of conjugacy classes instead. Lemma 4. If χ := χ47A + χ2A then χ(g) > 0 for every g ∈ B. Proof. Let g ∈ B. The group B has only one class of cyclic subgroups of order 47 so if o(g) = 47 then we have χ(g) > 0. Suppose o(g) = 47. Structure constant calculations show that every involution centralizer contains elements of class 2A and so if g is a power of an element of even order we must have χ(g) > 0. The only elements that have yet to be dealt with have order 31. Any such element is contained in a maximal subgroup with structure [230 ]L5 (2) which can also be shown to contain element of class 2A. Combining the above with the natural generalisation of Lemma 2 provides the upper bound given in Table 2. Note that we cannot prove the above in the same computational manner as the results of the previous sections since GAP does not contain all the primitive permutation characters of B in its libraries. We further note that only one class of maximal subgroups contains elements of order 47 - copies of the Frobenious group 47:23. To obtain a result like the above we must therefore use either class 23A or class 5 47A. Using 23A proves an upper bound that is worse than the best previously known upper bound. The above result is therefore the best possible. 4.2. The Monster. The Monster group M requires special attention the methods of previous sections cannot be applied as easily in this case since, at the time of writing, the maximal subgroups of M, and thus the primitive permutation characters of M, have yet to be classified. All not lost! We can still find a support class in this case using information about its conjugacy classes and the known maximal subgroups. Lemma 5. Class 2B of M is a support class. Proof. We aim to show that the primitive permutation characters defined by the known maximal subgroups of M are sufficient to give us χ2B (g) > 0 for every g ∈ M. First note that M has only two classes of involutions (see the character table given in [10, p.220]) and so the centralizer of any involution contains 2B elements. Both the 2A and 2B centralizers are known to be maximal [10, p.228] and so the sum defining χ2B must contain both of the permutation characters corresponding to these subgroups. Now, let g ∈ M and suppose there exist elements h, k ∈ M k = 1 such that g a = hb = k and hc is in class 2B for some a, b, c ∈ Z+ . Then g, h ∈ CM (k), which must contain a 2B element. It follows that the sum defining χ2B must contain the permutation characters defined by any maximal subgroups containing CM (k). (For instance, if g is in class 119A then we can find an h in class 14B such that k := h2 = g 17 which is in class 7A, so a = 17, b = 2 and c = 7 in this case. A maximal subgroup containing a 7A centralizer will therefore contain elements of class 119A and 2B and so the sum defining χ2B must contain the permutation character corresponding to such a subgroup.) Finally, from the fusion maps in the character table we see that the only classes not yet accounted for are the elements of orders 41, 59 and 71. It is known that M contains maximal copies of 41:40, L2 (59) and L2 (71) (see for instance [5, Table 1]). Furthermore, it is well known that the product of any two 2A elements of M has order at most 6. Since each of these subgroups only contain one class of involutions and in each of these subgroups there is a pair of involutions whose product is greater than 6, they must each contain 2B elements. It follows that the sum defining χ2B must contain the permutation characters defined by each of these classes of maximal subgroups. We thus have that χ2B (g) > 0 for all g ∈ M, so 2B is a support class by Lemma 3. Note we cannot replace 2B by 2A and improve this bound as this would give an ‘upper bound’ less than the lower bound of [4]. 6 5. Concluding remarks (1) Our first remark is clear: it would be of great interest to obtain better bounds on the spreads of simple groups, if not determine them all precisely. In particular, as noted in the MathsciNet review of [3], the bounds 3 ≤ s(M12 ) ≤ 9 are tantalizingly close and since M12 is such a relatively small and low degree permutation group it seems likely that this particular case is unusually within reach. (2) As Table 3 shows, support classes behave very erratically posing several questions regarding their nature. Which groups possess support classes? Is class 2A a support class infinitely often and if so, for which groups is it a support class? Conversely, among the groups for which 2A is not a support class which other classes are support classes? Are there groups whose smallest support class has order greater then 3 (class 3A is the smallest support class for the sporadic Suzuki group) and if so, how large can the order of such a class get? Are there infinitely many groups with a support class of elements of order 3? Of order 4? Of order 5? etc. It would be of great interest to see answers to all of these questions. (3) There is a more restricted notion of uniform spread, where we require the mates of our sets to lie in a single conjugacy class of G independent of the choice of the elements of the sets. In general, spread and uniform spread need not coincide: the group SL3 (2) has uniform spread exactly 3 but exact spread 4. It would be of great interest to determine, or at least bound, the uniform spreads of the sporadic simple groups, which has received much less interest in the literature than exact spreads have [8, 9]. Clearly the uniform spread is at most the exact spread and so upper bounds, like those proved here, also provide upper bounds on the uniform spread. Otherwise, the only known bounds on the uniform spreads of the sporadic simple groups, as far as the author is aware, are as follows: the uniform spreads of M11 and M12 are both 3 [7, Sections 5.9 and 5.10]. 6. Acknowledgments The author wishes to express his deepest gratitude to Professor Jamshid Moori for introducing him to the concept of spread in the first place and to the referee for making suggestions that resulted in substantial improvements to this paper. I am also grateful to Professor Robert Wilson for helpful correspondence regarding the cases of the Baby Monster and Monster groups. 7 References [1] M. Aschbacher and R.M. 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Wilson “The Maximal Subgroups of the Baby Monster, I”, J. Algebra, 211, (1999), 1–14 [18] A. Woldar “The exact spread of the Mathieu group M11 ”, J. Group Theory, 10, (2007), 167-171 Department of Economics, Mathematics and Statistics, Birkbeck, University of London, Malet Street, London WC1E 7HX E-mail address: bfairbairn@ems.bbk.ac.uk 8