A Recognition Algorithm for Special Linear Groups
نویسندگان
چکیده
At the 1988 Oberwolfach Computational Group Theory meeting, Joachim Neubiiser asked for an efficient algorithm to decide whether the subgroup of the general linear group GL(d, q) generated by a given set X of non-singular d X d matrices over a finite field ¥q contains the special linear group SL(rf, q). Amplifying the problem he said he would be happy to have a program that could be used for general purpose practical computation in the range rf ^ 30; Charles Sims suggested that one should be more ambitious as to degree (say d ^ 70) but restrict oneself to the case q = 2. In this paper we shall propose a method which we believe to be practical for values of d up to somewhere around 60 or 70 and which is not particularly sensitive to the size of q. We must emphasize however that our ideas have not yet, at the time of writing, been implemented and tested. Until adequate evidence has accumulated such claims must not be taken very seriously. Our algorithm is partly probabilistic. When its response is positive it tells us that the group generated by X certainly does contain SL(rf, q), but if the output is negative, that assertion will usually only be made with probability 1 — e, where e is a pre-determined small positive number. The analogous problem for permutation groups is to decide, for a given set X of permutations of a set of size n, whether the subgroup generated by X contains the alternating group Alt(n). An elegant probabilistic algorithm for recognizing alternating and symmetric groups has been in use for many years. Let us call an element of the symmetric group Sym(n) prima if one of its cycles has prime length p, none of the other cycles have length divisible by p, and p =s n — 3. The method exploits the following facts:
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تاریخ انتشار 1992