Constructing Permutation Representations for Matrix Groups

Abstract

New techniques, both theoretical and practical, are presented for constructing permutation representations for computing with matrix groups defined over finite fields. The permutation representation is constructed on a conjugacy class of subgroups of prime order. We construct a base for the permutation representation, which in turn simplifies the computation of a strong generating set. In addition, we present an elementary test for checking the simplicity of the permutation image.The theory has been successfully tested on a representation of the sporadic simple group Ly, discovered by Lyons (1972). With no a priori assumptions, we find a permutation representation of degree 9 606 125 on a conjugacy class of subgroups of order 3, find the order of the resulting permutation group, and verify simplicity. A Monte Carlo variation of the algorithm was used to achieve better space and time efficiency. The construction of the permutation representation required four CPU days on a SPARCserver 670MP with 64 MB. The permutation representation was used implicitly in the sense that the group element was stored as a matrix, and its permutation action on a "point" was determined using a pre-computed data structure. Thus, additional computations required little additional space. The algorithm has also been implemented using the MasPar MP-1 SIMD parallel computer and 8 SPARC-2's running under MPI. The results of those parallel experiments are briefly reviewed.

A parallel architecture for disk-based computing over the Baby Monster and other large finite simple groups

“…In 1994, Cooperman, Finkelstein, Tselman and York [15,16] constructed a permutation representation of Lyons's group of degree 9,606,125 from matrix generators, and also produced a strong generating set. Their representation was of permutation degree 9,606,125 for Lyons's group acting on a conjugacy class of subgroups of order three.…”

“…Weller [38,39] also produced a permutation representation of Janko's J4 group, using some of the hashing techniques of [15,16] and the double coset trick of [23,24]. That work was used in a revised existence proof for Janko's group [19].…”

“…This method was developed by Cooperman et al [15,16], and the original papers describe details, and additional optimizations not discussed here. We begin by considering the conjugate action of the group on subgroups of prime order.…”

“…This is accomplished by choosing at random an initial vector v of the underlying vector space. It is shown [15,Lemma 3.1] that with high probability, the randomly chosen initial vector v has the property that a group element g in the group is uniquely determined by the image vector vg. (The probability of having chosen a "bad" v is less than |G|/q n−m for a group G ≤ GL(n, q) with m the maximum dimension of a fixed point subspace of any non-identity element of G.…”

“…Some major achievements have been reached, such as the first constructions of permutation representations and strong generating sets for the Lyons group [15] and Janko's group J4 [19], and now even the Thompson group [38,39,40,20]. These accomplishments have been helped along by the sharing of information at websites such as Wilson's Atlas Web Page [41] which provides initial matrix representations for standard generators.…”