Generating finite soluble groups

Generating Sequences of Finite Groups

Irredundant Generating Sets of Finite Nilpotent Groups

It is a standard fact of linear algebra that all bases of a vector space have the same cardinality, namely the dimension of the vector space over its base field. If we treat a vector space as an additive abelian group, then this is equivalent to saying that an irredundant generating set for a vector space must have cardinality equal to the dimension of the vector space. The same is not true for...

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...

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...

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...