Generating random elements of a finite group

Abstract

We present a "practical" algorithm to construct random elements of a finite group. We analyse its theoretical behaviour and prove that asymptotically it produces uniformly distributed tuples of elements. We discuss tests to assess its effectiveness and use these to decide when its results are acceptable for some matrix groups.

The Ore conjecture

“…(1 + t 2i−1 ) 4 /(1 − qt 2i ), which is    q 5 + 7q 4 + 25q 3 + 69q 2 + 146q + 172, n = 5, q 6 + 7q 5 + 25q 4 + 70q 3 + 171q 2 + 318q + 325, n = 6, q 7 + 7q 6 + 25q 5 + 70q 4 + 178q 3 + 387q 2 + 643q + 594, n = 7.…”

“…This implies that every element of a large finite simple group is a product of two commutators. In [19] it is shown that the commutator map on finite simple groups is almost measure-preserving, a result having applications to the product replacement algorithm [7].…”

The Alperin and Uno Conjectures for the Fischer Simple Group Fi 22

“…Its maximal subgroups were constructed using the details supplied in [9] and the black-box algorithms of Wilson [20]. We also made extensive use of the algorithm described in [10] to construct random elements, and the procedures described in [3] and [4] The computations reported in this paper were carried out using Magma V.2.6-2 on a Sun UltraSPARC Enterprise 4000 server.…”

Recognition of Small Dimensional Representations of General Linear Groups

“…We use the algorithm of [6] to generate random elements and in step 1 choose a sample of size 4d 2 .…”

“…Another, more practical, option is the product replacement algorithm of Celler et al [6], which also runs in polynomial time (see [16]). For a discussion of both algorithms, we refer the reader to [18, pp.…”