Generating Random Elements in Finite Groups

Finite Groups With a Certain Number of Elements Pairwise Generating a Non-Nilpotent Subgroup

Random walks on generating sets for finite groups

We analyze a certain random walk on the cartesian product G of a finite group G which is often used for generating random elements from G. In particular, we show that the mixing time of the walk is at most crn 2 logn where the constant cr depends only on the order r of G.

Generating Random Elements of a Finite Group

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.

Generating random elements of abelian groups

Algorithms based on rapidly mixing Markov chains are discussed to produce nearly uniformly distributed random elements in abelian groups of finite order. Let A be an abelian group generated by set S. Then one can generate ε-nearly uniform random elements of A using 4|S| log(|A| /ε) log(|A|) additions and the same number of random bits.

The Schreier-Sims algorithm

A base and strong generating set provides an effective computer representation for a permutation group. This representation helps us to calculate the group order, list the group elements, generate random elements, test for group membership and store group elements efficiently. Knowledge of a base and strong generating set essentially reduces these tasks to the calculation of certain orbits. Giv...

Connectivity of the Product Replacement Graph of Simple Groups of Bounded Lie Rank

The Product Replacement Algorithm is a practical algorithm for generating random elements of a finite group. The algorithm can be described as a random walk on a graph whose vertices are the generating k-tuples of the group (for a fixed k). We show that there is a function c(r) such that for any finite simple group of Lie type, with Lie rank r, the Product Replacement Graph of the generating k-...