Generating sets of elements in compact groups

Generating Sets of Mathieu Groups

Julius Whiston [6] calculated the maximum size of an irredundant generating set for Sn and An by examination of maximal subgroups. Using analogous considerations, we will compute upper bounds to this value for the first two Mathieu groups, M11 and M12. Computational results gave explicit irredundant generating sets of M11 and M12 of size 5 and 6, respectively. Together these give the full resul...

Almost commuting elements in compact Lie 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.

Properties of Generating Sets of Finite Groups

Students in the project should have a solid background in basic linear algebra and abstract algebra. For example, the first six chapters of the textbook by Dummit and Foote, Abstract Algebra, 3rd edition, Wiley, would be more than adequate for the required background in group theory. For example, section 5.2 defines the term elementary divisors which appears below and in one standard descriptio...