Automorphism groups of resolvable incidence structure

Author

Grant Harlan Lathrom

Keywords and Phrases

Resolvable incidence structures; Frequency hypercubes

Abstract

A net is a resolvable incidence structure which can be thought of as generalizing the points and lines of an affine plane. Well known examples of nets include the Affine Resolvable Balanced Incomplete Block Designs which arise from complete sets of mutually orthogonal Latin squares. The blocks of a net can be partitioned into equivalence classes called parallel classes, in which no two blocks in the same parallel class meet, and any two blocks from distinct parallel classes meet in exactly the same number of points.

An automorphism of a net is a bijection on the set of points of the net which preserves the incidence structure between the points and blocks of the net. In the case of two parallel classes, we will provide two characterizations of the full auto­ morphism groups of the nets. One characterization gives the automorphism group in terms of the wreath product and semi-direct products of permutation groups. In the second characterization, we will give the full automorphism group in terms of the amalgamated product of incidence substructures which are induced naturally from the original net. We then generalize the notion of a net by relaxing the condition that any two blocks from distinct parallel classes meet in the same number of points. Under certain conditions on the cardinality of these intersections, we are again able to give a characterization of the full automorphism group of these generalized nets.

Finally, in the case of more than two parallel classes, we introduce a function on the indexing set of the blocks and parallel classes called the weighting function. Using a set-valued function defined on a permutation group, we provide a characterization of the full automorphism group of a net, when the weighting function is sufficiently symmetric, in terms of objects which depend on the intersection properties of the design, rather than any underlying algebraic structure used to generate the design--Abstract, p. iii

Advisor(s)

Ilene H. Morgan

Committee Member(s)

Louis Grimm
Miron Bekker
Gary L. Gadbury
William Weeks, IV

Department(s)

Mathematics and Statistics

Degree Name

Ph. D. in Mathematics

Publisher

University of Missouri--Rolla

Publication Date

Fall 2003

Pagination

vii, 119 pages

Note about bibliography

Includes bibliographical references (pages 117-118).

Rights

© 2003 Gorman Lathrom, All rights reserved.

Document Type

Dissertation - Restricted Access

File Type

text

Language

English

Subject Headings

Automorphisms

Thesis Number

T 8371

Print OCLC #

56363715

Recommended Citation

Lathrom, Grant Harlan, "Automorphism groups of resolvable incidence structure" (2003). Doctoral Dissertations. 1516.
https://scholarsmine.mst.edu/doctoral_dissertations/1516