A Recognition Algorithm for Special Linear Groups

Abstract

contain th speciase l linea SL(rfr grou, q).pAmplifying the problem he said he would be happy to have a program that couldbe used fo r genera l purpose practical computatio in th range e rf ^n 30; Charle sSims suggested tha onte should be more ambitiou ass to degree (sa y d ^ 70 ) bu trestrict onesel tfo the case q = 2. In this pape wr e shall propos ae method whichwe believ teo be practica fol r value osf d up t somewhero e aroun 60d or 70 andwhich is no particularlt y sensitiv to thee size of q. W muse t emphasize howeverthat our ideas hav noe t yet, at th time e of writing, been implemente andd tested.Until adequate evidenc ha accumulates e d such claim nost mus b takee t n veryseriously. Ou algorithr ims partly probabilistic it. Whes respons n ies positiv iettells us that the group generate bdy X certainly does contai SL(rfn , q), but if theoutput is negative, that assertion will usually b onle mady e with probabilit 1y — e,where e is a pre-determined small positive number.The analogous proble fomr permutatio n group is tso decide fo, r a given se t Xof permutation osf a set o sizf e n, whethe thr e subgroup generate bdy X containsthe alternating group Alt(n) An elegan. t probabilistic algorith fo recognizinr m galternating and symmetric group hass been in use fo manr y years Le. t u cals l anelement of th symmetrie c group Sym(n if) on primae of cycle its has prims elength p, non oef the othe r cycles have length divisibl by p, ane d = ps n — 3. Themethod exploit thse following facts:(i) if G i Alt(/zs ) or Sym(n) the an good proportio onf th elemente s o f G areprima;(ii) if G does no contait n Alt(n) and, if G i primitivs a e permutation groupthen G contains no such elements.The forme irs no hart d to prove (see , fo exampler [20, Lectur, e 11]); th lattee rassertion comes from a well-known old theorem of Jordan (see, fo exampler [30, ,Theorem 13.9]). Accordingl the ide ya i s to tes it f G i primitives , whic cahn bedone pretty quickl byy, fo exampler a, method of Atkinson I.f no thet n certainlyG is neither Alt(n no) Sym(n)r I.f G i primitives th, e crucial ste isp to make asuitable numbe orf independent random selection of elements osf G. No mant yare needed—see fo,r exampl [20]e . som If e prima permutation show up thens Gis identified correctl as Alt(ny) o r Sym(n) I.f not the, n with high probabilit G ydoes not contain Alt(n). Thus ther is a smale l probability tha the algoritht m willfail to identify Alt(n or Sym(n) ) . This procedur wa firsts e programme ford