Generating Random Elements in Finite Groups

Generating Random Elements in Finite Groups

Let G be a finite group of order g. A probability distribution Z on G is called ε-uniform if |Z(x) − 1/g| ≤ ε/g for each x ∈ G. If x1, x2, . . . , xm is a list of elements of G, then the random cube Zm := Cube(x1, . . . , xm) is the probability distribution where Zm(y) is proportional to the number of ways in which y can be written as a product x1 1 x ε2 2 · · · xεm m with each εi = 0 or 1. Let...

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.

Generating random elements of finite distributive lattices

This survey article describes a method for choosing uniformly at random from any finite set whose objects can be viewed as constituting a distributive lattice. The method is based on ideas of the author and David Wilson for using “coupling from the past” to remove initialization bias from Monte Carlo randomization. The article describes several applications to specific kinds of combinatorial ob...

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.

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