On the Efficiency of Some Finite Groups

Abstract

We describe a new technique for finding efficient presentations for finite groups. We use it to answer three previously unresolved questions about the efficiency of group and semigroup presentations.

Nice Efficient Presentions for all Small Simple Groups and their Covers

“…The availability of systems for computational group theory (for example, GAP [14], Magma [2] and Magnus [21]) makes it quite easy to experiment with groups. Havas, Newman and O'Brien [19] have developed a Magma program that enables us to find all distinct generating sets for moderately sized permutation groups. (The program uses representatives from appropriately merged orbits of the action of the automorphism group of each permutation group studied.)…”

“…This group has been considered in detail by Havas, Newman and O'Brien [19] in the context of efficient semigroup presentations. For U 3 (3), we investigated 1442 distinct generating pairs, and found two efficient presentations amongst presentations on these generating sets.…”

On presentations for unitary groups

“…• Generating sets. We consider generating pairs for groups based on matrix or permutation representations, sometimes using complete sets [14] to facilitate the process. We can construct presentations on pairs with specific properties.…”

“…For q ≤ 5 we compute complete sets of generating pairs, as in [14], but this is not well suited to larger groups. We reduce the number that we examine in various ways.…”

“…The group U 3 (3) is discussed in [14] and [3]. Four efficient presentations are listed in [3], including the (4, 12)-generated one of Table 2.…”