Group decompositions, Jordan algebras, and algorithms for p-groups
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Date
2008-06
Authors
Wilson, James B., 1980-
Journal Title
Journal ISSN
Volume Title
Publisher
University of Oregon
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 semi-simple 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 polynomial-time 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.
Description
viii, 125 p. A print copy of this thesis is available through the UO Libraries. Search the library catalog for the location and call number.
Keywords
Computer science, Mathematics, p-groups, Jordan algebras, Group decompositions, Central products, Direct products, Algorithms