M.Sc Thesis
M.Sc Student Elad Doron Enumerating Semiabelian p-Groups Department of Mathematics Professor Jack Sonn

Abstract

The notion of a "semiabelian group" was introduced by Dentzer, who proved that for any base field k, all semiabelian groups occur as Galois groups of a geometric Galois extension of the rational function field k(t).

Using GAP computer system, Dentzer also made a rough classification of a special subfamily of the non-semiabelian groups called- "irreducible non-semiabelian groups", of order up to 100.

Independently, Kisilevsky and Sonn have solved the minimal ramification problem for a family Gp of p-groups.

Following Dentzer's and Kisilevsky's and Sonn's work, Neftin proved that the family Gp, for which the minimal ramification problem is solvable, actually coincides with the family of semiabelian p-groups, which we will denote from now on by SA.

Kisilevsky's and Sonn's result, regarding the solubility of the minimal ramification problem in groups lying in the family SA, raises the following natural question: how many groups lie inside SA ? Since there is an infinite amount of groups, the actual analogous question is- what is the density of the family SA inside the entire family of p-groups P, where the notion of "density" is considered in a limit context.

In this work, we first aim to find a possible way of partitioning the families SA,P in a way that will allow us to estimate the density of the family SA inside the entire family P. We used the Frattini series {Phii(Fp)} of a free pro-p group Fp in order to partition the mentioned families. Namely, we found bounds for the number of normal subgroups S of a free pro-p group Fp of rank r for which the quotient Fp/S is a semiabelian p-group, and Phin(Fp) is a subset of S.

We prove that using this partition, the density of the family of semiabelian p-groups inside the family of p-groups approaches zero when taking n to approach infinity.