Nice Efficient Presentions for all Small Simple Groups and their Covers

Abstract

Prior to this paper, all small simple groups were known to be efficient, but the status of four of their covering groups was unknown. Nice, efficient presentations are provided in this paper for all of these groups, resolving the previously unknown cases. The authors‘presentations are better than those that were previously available, in terms of both length and computational properties. In many cases, these presentations have minimal possible length. The results presented here are based on major amounts of computation. Substantial use is made of systems for computational group theory and, in partic-ular, of computer implementations of coset enumeration. To assist in reducing the number of relators, theorems are provided to enable the amalgamation of power relations in certain presentations. The paper concludes with a selection of unsolved problems about efficient presentations for simple groups and their covers.

Presentations of finite simple groups: a computational approach

“…Additional fine-tuning when 4 ≤ n ≤ 8: In these cases there is a presentation Y | S for T = A n with 2 generators and 3 relations [ThS,CRKMW,CHRR1]. Now we obtain 4 + 2 generators and 14 + 3 + 5 relations.…”

“…(Havas found this using [CHRR1,Method 2], modified to handle groups that are not necessarily perfect.) This time ρ = 1 and |X 1 | = 2, so that S 12 has a presentation with 2 generators and 2 + 1 + 2 relations.…”

“…Finally, we leave SU(3, 2) = 3 1+2 Q 8 to the reader, SU(3, 3) is in [CHRR1], and the case SU(3, 5) follows from the efficient presentation for PSU(3, 5) in [CHRR2] together with Lemma 2.2.…”

“…If X | R is a presentation of a finite group G, then |R| − |X| is at least the smallest number d(M) of generators of the Schur multiplier M of G; and X | R is called an efficient presentation if |R| − |X| = d(M) [CRKMW,CHRR1,CHRR2]. For 35 years the groups PSL(2, p) with p ≥ 5 prime have contained the only infinite family of finite nonabelian simple groups known to have efficient presentations ( [Sun,CR3]; cf.…”

Short presentations for alternating and symmetric groups

“…In particular, we have the following consequence of Theorems 1.1 and 1.2. [7] about the existence of a 2-generator presentation for A n with a bounded number of relators.…”