Group decompositions, Jordan algebras, and algorithms for p-groups
| dc.contributor.author | Wilson, James B., 1980- | |
| dc.date.accessioned | 2009-01-15T00:44:03Z | |
| dc.date.available | 2009-01-15T00:44:03Z | |
| dc.date.issued | 2008-06 | |
| dc.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. | en |
| dc.description.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. | en |
| dc.description.sponsorship | Adviser: William M. Kantor | en |
| dc.identifier.uri | https://hdl.handle.net/1794/8302 | |
| dc.language.iso | en_US | en |
| dc.publisher | University of Oregon | en |
| dc.relation.ispartofseries | University of Oregon theses, Dept. of Mathematics, Ph. D., 2008; | |
| dc.subject | Computer science | en |
| dc.subject | Mathematics | en |
| dc.subject | p-groups | en |
| dc.subject | Jordan algebras | en |
| dc.subject | Group decompositions | en |
| dc.subject | Central products | en |
| dc.subject | Direct products | en |
| dc.subject | Algorithms | en |
| dc.title | Group decompositions, Jordan algebras, and algorithms for p-groups | en |
| dc.type | Thesis | en |
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