For each family of finite classical groups, and their associated simple quotients, we provide an explicit presentation on a specific generating set of size at most 8. Since there exist efficient algorithms to construct this generating set in any copy of the group, our presentations can be used to verify claimed isomorphisms between representations of the classical group. The presentations are available in Magma.
Rarely do we know explicit words (short or otherwise) to express standard generators in terms of the generators used in @cite_10 , or vice versa, so it is not feasible to convert their presentations to ones on standard generators. One significant obstruction is that sometimes generators used there are identified only by specifying properties they must satisfy. Nor do constructive recognition algorithms employing their generators exist for classical groups, so these presentations cannot be used directly to verify the necessary isomorphisms. Theorem 10.1 of @cite_10 illustrates an additional concern: it employs a generator @math specified only up to certain properties. This element does not exist in @math ; in private communication in 2012, the authors of @cite_10 fixed the error. Thus, while the results of @cite_18 @cite_19 provide spectacular answers to long-standing challenging problems, we believe that our presentations are necessary for our significant algorithmic application.
@cite_19 @cite_18 @cite_10
2123560844 1997911818
There is a constant such that all nonabelian finite simple groups of rank over , with the possible exception of the Ree groups , have presentations with at most generators and relations and total length at most . As a corollary, we deduce a conjecture of Holt: there is a constant such that for every finite simple group , every prime and every irreducible -module All nonabelian finite simple groups of rank @math over a field of size @math , with the possible exception of the Ree groups @math , have presentations with at most @math relations and bit-length @math . Moreover, @math and @math have presentations with 3 generators @math 7 relations and bit-length @math , while @math has a presentation with 7 generators, @math relations and bit-length @math