Abstract:

Finite p groups are studied using bilinear methods which lead to using nonassociative rings. There are three main results, two which apply only to p groups and the third which applies to all groups.
First, for finite p groups P of class 2 and exponent p the following are invariants of fully refined central decompositions of P : the number of members in the decomposition, the multiset of orders of the members, and the multiset of orders of their centers. Unlike for direct product decompositions, Aut P is not always transitive on the set of fully refined central decompositions, and the number of orbits can in fact be any positive integer. The proofs use the standard semisimple and radical structure of Jordan algebras. These algebras also produce useful criteria for a p group to be centrally indecomposable.
In the second result, an algorithm is given to find a fully refined central decomposition of a finite p group of class 2. The number of algebraic operations used by the algorithm is bounded by a polynomial in the log of the size of the group. The algorithm uses a Las Vegas probabilistic algorithm to compute the structure of a finite ring and the Las Vegas MeatAxe is also used. However, when p is small, the probabilistic methods can be replaced by deterministic polynomialtime algorithms.
The final result is a polynomial time algorithm which, given a group of permutations, matrices, or a polycyclic presentation; returns a Remak decomposition of the group: a fully refined direct decomposition. The method uses group varieties to reduce to the case of p groups of class 2. Bilinear and ring theory methods are employed there to complete the process. 