# On O-Basis Groups and Generalizations

## Date

2007-08-15## Type of Degree

Dissertation## Department

Mathematics and Statistics

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Show full item record## Abstract

A class of finite groups which we call o-basis groups is generalized and explored. One reason for interest in these groups lies with the concept's origins. The notion of o-basis group arose from the study of the existence, in the n-fold tensor product of a complex inner product space, of an orthogonal basis consisting entirely of ""standard symmetrized tensors"". We call such a basis an o-basis. The term ""symmetrized"" refers to the action on the tensor product of a subgroup of the symmetric group Sn. Given a subgroup of Sn, one may ask if the corresponding symmetrized tensor space has an o-basis. The answer will depend in part on the structure of the given group. Since any group can be homomorphically embedded onto a subgroup of the symmetric group, arbitrary finite groups may be considered. It has already been shown that if G is an o-basis group and f from G into Sn is a homomorphism, then the symmetrized space corresponding to f(G) has an o-basis. The study of these groups therefore may well be of interest to those working with o-bases of symmetrized spaces. Our focus, however, is on the group structure and character theory of o-basis groups themselves with a view toward using the o-basis property as a means of distinguishing between abstract finite groups. The tools come from finite group theory and the character theory of finite groups. Field theory appears very briefly. In previous work, some interesting classes of groups have been shown to be o-basis, and so far all groups identified as o-basis are nilpotent. Particularly compelling are the dihedral groups. It has been shown that the o-basis dihedral groups are precisely those that are 2-groups. These are also precisely the nilpotent dihedrals. With this in mind, we ask whether or not all o-basis groups are nilpotent. We consider this question for a restricted class of groups. Conversely, there are examples of nilpotent groups that are not o-basis leading us to explore conditions on a nilpotent group which will guarantee that the group is o-basis. The results obtained indicate a possible connection between the o-basis property and the nilpotence class of a group. The second main division of the present work is an exploration of a generalization of o-basis groups. While the following definitions contain technicalities, the reader should be able, without preliminary preparation, to understand the nature of the generalization. A group is o-basis if for each subgroup H of G and irrecucible character f of G for which (f, 1)_H is non-zero, there are a certain number of ""orthogonal cosets"" of H in G. We generalize by relaxing the subgroup condition as follows. Let K be a subgroup of G. We say G is K-o-basis if for each irreducible character of G and each subgroup H containing K where (f, 1)_H is non-zero, there are the required number of orthogonal cosets of H. The o-basis groups, therefore, are the E-o-basis groups, where E denotes the identity subgroup. Note that to apply this notion to a given class of groups, K must be defined for all groups in that class. Some results are obtained for the case when K is a member of the lower central series and when it is a member of the upper central series. Finally, the still open question of whether a direct product of o-basis groups is o-basis is brie°y discussed.