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