The perfect groups of order up to two million

Abstract

We enumerate the 15768 perfect groups of order up to 2·10, up to isomorphism, thus also completing the missing cases in [HP89]. The work supplements the by now wellunderstood computer classifications of solvable groups, illustrating scope and feasibility of the enumeration process for nonsolvable groups. The algorithmic setup for constructing finite groups of a given order, up to isomorphism, has been well-established, both in theory and in practice, for the construction of groups [BEO02, EHH17]. It proceeds inductively, by constructing extensions of known groups of smaller orders and eliminating isomorphic candidates when they arise. Due to limitations in implementations of underlying routines, this however had been done so far mostly for solvable groups. The aim of this paper is to show the feasibility of generalizing this approach to the case of nonsolvable groups. Instrumental in this has been the calculation of 2-cohomology through confluent rewriting systems, generalizing the method [HEO05, §8.7.2] for solvable groups that uses a PC presentation. The construction process is illustrated by revisiting the enumeration of perfect groups that was started in [HP89] and to extend it to order 2 ·10. In total we find 15768 perfect groups, seeded from the 66 nonabelian simple groups of order up to 2 · 10. Compared with [HP89], this newly provides explicit lists of the groups of orders 61440, 86016, 122880, 172032, 245760, 344064, 368640, 491520, 688128, 737280, 983040 that were omitted in their classification of groups of order up to 10. In this range, the calculations also found five groups (in addition to two groups found already in 2005 by Jack Schmidt) that had been overlooked in [HP89]. Besides serving as examples for testing conjectures, such lists of groups are used as seed in algorithms for the calculation of subgroups of a given finite group [Neu60, CCH01, Hul13], or indeed for the construction of all groups of a given order. All calculations were done using the system GAP [GAP20], which also serves as repository of the resulting group data. The program that performed the classification is available at https://github.com/hulpke/perfect and should allow for easy generalization or extension. In addition to the actual classification result, this work also serves as a prototype of enumeration of nonsolvable groups, extending the work of [BEO02] to the nonsolvable case. It illustrates the feasibility range of current implementations of underlying routines for cohomology, extensions, and isomorphism tests, with a number of general-purpose improvements in the system GAP [GAP20] (that will be part of the 4.12 release) by the author having been motivated by this work. Indeed, the fact, that it took over 30 years since the publication of [HP89] to complete the classification of perfect groups up to order one million, indicates the broad infrastructural 1 ar X iv :2 10 4. 10 82 8v 3 [ m at h. G R ] 6 J ul 2 02 1 requirements of such classifications, with isomorphism tests [CH03] being the most prominent utility (and ultimately the bottleneck of any classification). 1. The construction process We first briefly summarize the construction process for perfect groups of a given order n > 1. This process closely follows the description in [HEO05, §11.3] (and, apart from seeding with nonabelian simple groups, is fundamentally the same strategy as used in [BEO02] for solvable groups). The construction of perfect groups for a chosen order n consists of two parts, depending on whether the resulting groups have a solvable normal subgroup or not. 1.1. Fitting-free groups. Groups that have no solvable normal subgroup are called Fittingfree. Such a group G embeds into the automorphism group of its socle S CG, which in turn is a direct product of simple nonabelian groups. The conjugation action of G on the k direct factors of S induces a permutation representation of G of degree k. For its image to be nontrivial perfect, we would need k ≥ 5 (and thus n = |G| ≥ 60). For the order range considered, this means that this image is trivial, thus all direct factors of S must be normal in G. But then G/S is isomorphic to a subgroup of the direct product of the automorphism groups of the simple nonabelian socle factors. Such a factor group is solvable (by the Schreier conjecture), showing that we must have G = S as a direct product of simple nonabelian groups. For n ≤ 2 · 10, the possible direct factors to consider are: A5, A6, A7,PSL(3, 3),PSU(3, 3),M11, A8,PSL(3, 4),PSp(4, 3), Sz(8), PSU(3, 4),M12,PSU(3, 5), J1, A9,PSL(3, 5),M22, J2,PSp(4, 4), A10,PSL3(7). and PSL(2, q) for prime powers 7 ≤ q ≤ 157, q 6= 128. (Note that PSL(2, 4) ∼= PSL(2,5) and PSL(2, 9) ∼= A6.) 1.2. Inductive construction. Groups of order n that possess a solvable normal subgroup can be constructed as extension of groups of smaller order d | n by a simple module of order p = n/d. As factor groups of perfect groups these smaller groups need to be perfect themselves. We thus assume that, by induction, all perfect groups of order dividing n are known. (Of course the existence of perfect groups of order d is only necessary, but not sufficient, for the existence of perfect groups of order n = p · d.) We also can assume that p | d if a = 1, since the action of a perfect group on a 1dimensional module must be trivial, and any extension for p = n/d and p coprime to d thus would be a direct product and thus not perfect. This gives the following construction process: (1) Iterate over all proper divisors d | n with n/d = p, such that a > 1 or p | d. Then iterate over all perfect groups F of order d: (2) Classify the irreducible a-dimensional F -modules M over Fp. For this, we use the Burnside-Brauer theorem, as described in [HEO05, §7.5.5], to classify all modules, and eliminate those of the wrong dimension. (Clearly it is sufficient to consider modules for the factor group F/Op(F ) by the largest normal p-subgroup. The index of the kernel of the module action is further bounded by |GLa(p)|, which can eliminate some small dimensions a > 1 for groups that have no small proper factors.)