a recognition algorithm for special linear groups

A RECOGNITION ALGORITHM FOR SPECIAL LINEAR GROUPS PETER M. NEUMANN and CHERYL E. PRAEGER [Received 6 March 1991—Revised 20 November 1991]

1. Introduction 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: (i) if G is Alt(/z) or Sym(n) then a good proportion of the elements of G are prima; (ii) if G does not contain Alt(n), and if G is a primitive permutation group then G contains no such elements. The former is not hard to prove (see, for example, [20, Lecture 11]); the latter assertion comes from a well-known old theorem of Jordan (see, for example, [30, Theorem 13.9]). Accordingly the idea is to test if G is primitive, which can be done pretty quickly by, for example, a method of Atkinson. If not then certainly G is neither Alt(n) nor Sym(n). If G is primitive, the crucial step is to make a suitable number of independent random selections of elements of G. Not many are needed—see, for example [20]. If some prima permutation shows up then G is identified correctly as Alt(n) or Sym(n). If not, then with high probability G does not contain Alt(n). Thus there is a small probability that the algorithm will fail to identify Alt(n) or Sym(n). This procedure was first programmed for 1991 Mathematics Subject Classification: 20-04, 20C40, 20C20. Proc. London Math. Soc. (3) 65 (1992) 555-603.

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machine computation by E. T. Parker and Paul J. Nikolai in 1958 (see [21]). Many variations and improvements to the basic idea are possible and a highly efficient modern version has been prepared by Peter Cameron and John Cannon

PI-

Our approach to the problem for matrix groups is in some ways similar. An element x of GL(d, q) is said to be irreducible if its characteristic polynomial is irreducible over F^. Equivalently, x is irreducible if it acts irreducibly on the standard d-dimensional vector space V over F9 (which we take to be the space Vj, of row-vectors, with matrices acting by right multiplication). For d^3 we shall say that a matrix y is nearly irreducible if its characteristic polynomial factorises as a product (t - k)g(t) where g(t) is irreducible in FJf]; equivalently, if V may be expressed as a direct sum Vx © V2 of _y-invariant subspaces such that V, is 1-dimensional and y acts irreducibly on V2. The relevance of such matrices is the following. On the one hand, if SL(d, q)^G =s GL(d, q) then a fairly large proportion of the elements of G are irreducible and a fairly large proportion are nearly irreducible. Therefore if examination of a suitable number of randomly chosen elements of G fails to produce an irreducible or a nearly irreducible element then we can be pretty sure that G does not contain SL(d, q). We shall make this more precise in §2 and show, for example, that failure after examination of 5-5 d matrices would tell us that the probability that G contains SL(rf, q) is less than 1/100. On the other hand, most groups G «£ Gh{d, q) that contain both irreducible and nearly irreducible elements contain SL(d, q). Furthermore, the few exceptions are quite easily recognisable computationally. They are all of one of the forms described in the following list. CATALOGUE 1. Types of subgroups G of GL(d, q) that may contain an irreducible element x and a nearly irreducible element y. We take d 5= 4 and q = pb where p is a prime number; Z denotes the subgroup of scalar matrices contained in G. (1) Any group G containing SL(d, q). (2) Monomial groups: G is conjugate to a subgroup of the monomial group GL(1, q) wr Sym(d), that is, G preserves a direct-sum decomposition V = Vx ©... © Vd, where dim Vt = l for all / and we may take K, to be ^-invariant. If such a group is to contain suitable elements x and y then each prime divisor of d(d — 1) must divide q — \. (3) Groups G conjugate to a subgroup of TL(l,qd). If such a group is to contain suitable elements x and y then d must be prime and q must be a primitive root modulo d. (4) Groups representable modulo scalars over subfields: G is conjugate to a subgroup of GL(d, pc). Z for some divisor c of b. If such a group is to contain suitable elements x, y then a must be coprime with d(d — 1), where a := b/c. (5) Nearly simple groups: there exists G(> «a G such that GJZ is a non-abelian simple group T (which is very much smaller than PSL(d, q)) and GIZ =£ Aut T; furthermore, Go is primitive and absolutely irreducible on V, and the representation is not realizable over any proper subfield of Ff/.

AN ALGORITHM FOR LINEAR GROUPS

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The restriction to d s= 4 in this catalogue is unimportant. For rf = 2we define an element y to be nearly irreducible if its characteristic polynomial has two distinct roots in F^. With this convention we can add one more item to the list: (6) Small cases: d is 2 or 3, and G is contained in the normaliser of an extraspecial d-group R of order d3. If d = 3 then q = 1 (mod 6), x has order 3 modulo scalars, y has order 2 or 4 modulo scalars, and G/R is isomorphic to a subgroup of the affine group ASL(2, 3). If d = 2 then q is odd and G/Z is Alt(4) or Sym(4).

We shall use a theorem of Aschbacher [1] to prove THEOREM 1. Let G be a subgroup of the general linear group GL(rf, q) that contains an irreducible element x and a nearly irreducible element y. Then G is of one of the types listed in Catalogue 1.

It is possible for groups in each of the classes (l)-(6) to contain both irreducible and nearly irreducible elements. A classification of such groups in the nearly simple case (5) could be used to produce improvements to our proposed algorithm. Unfortunately, that classification is still incomplete and we have found it necessary to work with certain slightly special kinds of irreducible and nearly irreducible elements in order to obtain a usable theorem. It was proved by Zsigmondy [31] in 1892 that, if a, d are integers, a ss 2, d ^ 3, and the pair (a, d) is not (2,6), then there is a prime which divides ad — 1 but does not divide a' - 1 for l^i^d — 1. Such a prime is called a primitive prime divisor of ad — 1. An irreducible matrix whose order is divisible by a primitive prime divisor of qd — 1 will be said to be primitive; similarly, we shall refer to a nearly irreducible matrix whose order is divisible by a primitive prime divisor of qd~l - 1 as being primitive. Suppose now that dsM and (d, q)^(6, 2), (7, 2). We shall verify in § 2 that the assertion made above for irreducible and nearly irreducible matrices still holds good for primitive matrices: that is, if SL(d, q)m G =s GL(d, q) then a fairly large proportion of the elements of G are primitive irreducible and a fairly large proportion are primitive nearly irreducible. Thus if we examine a suitable number (still about 5-5 d) of randomly chosen members of our group G and do not come up with such elements then we can be pretty sure (with probability greater than 0-99) that G does not contain SL(d, q). On the other hand, if we do find both a primitive irreducible element x and a primitive nearly irreducible element y then we are in a much stronger position. For, the subgroups of GL(rf, q) that contain elements of both types may be classified explicitly. We have done that in Theorem 2 below. We understand from Christoph Hering (letter dated 25 January 1990) that he has proved a very similar theorem. John H. Walter is working towards a more general theorem in a series of papers the first of which has appeared as [28]. Another more general study has been published (as was brought to our attention after this work was complete) also by Dempwolff [4], whose methods and applications are, however, completely different from ours. The list is as follows. CATALOGUE 2. List of nearly simple subgroups G ofGL(d, q), where d 2= 4, that contain a primitive irreducible element x and a primitive nearly irreducible element y. Thus T^G/Z^ Aut T where T is simple and Z is the group of scalar matrices contained in G; furthermore, G is primitive and absolutely irreducible, and the

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representation is not realizable modulo scalars over any proper subfield of \fq. As before, q =pb where p is a prime number. Further, r is a primitive prime divisor of the order of x and s is a primitive prime divisor of the order of y. (i) G = Z XT, where T = PSL(2, r), d = \(r — l) = s, q is a primitive root modulo d and q has order d modulo r. (ii) G is M23 or M24 in GL(11,2), r = 23 and s = 11. (iii) G = ZxT< GL(5, 3) where T = Mu and r = 11, s = 5. (iv) G/Z = Alt(7) and d = A, r = 5, s = 7. Here b = 1, that is, q is prime. Furthermore, either 2 then q is a primitive root modulo k. (iii) // the polynomial tk~l + utk~2 + ... + uk~2t + uk~* is nearly irreducible in ¥q[t) then k is 3 or 4.

Proof, (i) Let r be a prime divisor of k. If u had an rth root u0 in ¥q then the polynomial tk/r — u0 would divide tk — u. Thus if tk — u is irreducible then u is not an rth power in ¥q. It follows that r divides q — 1 and that the r-part of q -1 divides ord(u), as claimed. Note that there are many examples of irreducible polynomials of this form. For example, t2 — 2, t8-2, t6 — 3 are irreducible over F3, F5 and F7 respectively. (ii) Let f{t) := tk~l + utk~2 +... + uk~2t + uk~l = (tk - uk)/(t - u) and suppose that/(/) is irreducible in ^q[t]. Clearly we may assume that k 5s 3. If m > 1 and m divides k then (tm - um)/(t - u) divides f(t) and hence m-k. Thus k is prime. The roots of f(t) in a splitting field are uB', where 6 is a primitive kth root of 1 and 1 «£ i"as k — 1 and so the splitting field oif(i) is ^q(9). On the one hand, this must have degree k — 1 since fit) is irreducible, but, on the other hand, its degree is the least positive integer m such that k divides qm — 1, that is, the order of q modulo k. Thus q must be a primitive root modulo k. (iii) Suppose now that fit) is nearly irreducible. As in Part (ii), if k has a divisor m satisfying \ is bilinear. Then it extends to a non-singular g-invariant bilinear form, which we also denote by , on V K. As it is non-singular, there exists / =s d — 1 such that w e have x(g)(v0, u,-) = {vog, Vig) = (f>(kovO) kM) = AoAf-0(uo, vi). Thus A0A, = x(g). Applying & we see that Xf = A0, that is, AQ2' = Ao. Since Ao generates ¥qd, we must have o2' = 1 and it follows that 2/ is a multiple of d. But 2/ < 2d, so in fact IX = d, that is, d is even. If g actually preserves the form, that is, %{g) = 1, then A(7' = A,, and, since Ao is an arbitrary eigenvalue of g, we see that the eigenvalues occur in inverse pairs. Now suppose that is unitary with respect to an involutory automorphism x of F 9 . Then q = r2 for some r, and o = p2 where p : a>-*ar for a e K. The restriction of p to ¥q is x and our form extends to a non-singular g-invariant p-sesquilinear form 0 on V K. If u, v e V then a n d SO t n e multiplier x(g) lies in F r . Since the form is non-singular, (vQ, v^^O for some /. Then x(g)(vo> vi) = Hvog, Vig) = 0(ApVo, A/U,-) = AoAf0(uo, v^, so that A0Af = x(g). Applying p 2 / + 1 we see that AfAf2l+2 = X(g)p = x(g)- Thus kf+2 = k0, that is, kf+' = k0. Since Ao generates K over F,,, it follows that the order d of o divides (and in fact is equal to) 2/ + 1. Thus d is odd, as stated. Proof of Proposition 3.8. Let 0 be a non-singular bilinear or unitary form on V. If GAut(0) contains an irreducible element x then Lemma 3.9 applies directly to tell us that if (j> is bilinear then d must be even, and if is unitary then d must

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be odd. Suppose that y is a nearly irreducible element in GAut(#), and let v be an eigenvector of y. If v were isotropic (that is,