Some topics in the theory of finite groups
<p>A regular 2-graph consists of a set andOmega; together with a (non-empty) set <em>t</em> of three-element subsets of andOmega; such that any two-element subset of andOmega; is contained in the same number of elements of <em>t</em>, any four-element subset of andOmeg...
| Main Authors: | , |
|---|---|
| Other Authors: | |
| Format: | Thesis |
| Language: | English |
| Published: |
1971
|
| _version_ | 1797099163024883712 |
|---|---|
| author | Taylor, D Taylor, Donald E. |
| author2 | Higman, G |
| author_facet | Higman, G Taylor, D Taylor, Donald E. |
| author_sort | Taylor, D |
| collection | OXFORD |
| description | <p>A regular 2-graph consists of a set andOmega; together with a (non-empty) set <em>t</em> of three-element subsets of andOmega; such that any two-element subset of andOmega; is contained in the same number of elements of <em>t</em>, any four-element subset of andOmega; contains an even number of elements of <em>t</em> and not every three-element subset of andOmega; is in <em>t</em>. These objects were introduced by G.andnbsp;Higman who used a regular 2-graph with 276 points to provide a combinatorial setting for the doubly transitive representation of Conway's sporadic simple group C<sub>3</sub>.</p> <p>In this thesis it is shown that regular 2-graphs are in one-one correspondence with equivalence classes of strong graphs (as defined by J.J.andnbsp;Seidel). Moreover, for each point of a regular 2-graph there is a natural way of defining a strongly regular graph on the remaining points. These graphical representations are used to obtain restrictions on the structure and on the parameters of a regular 2-graph. It is also possible, via the strong graphs, to represent a regular 2-graph as a configuration of equiangular lines in Euclidean space. Conversely, results about regular 2-graphs obtained in this thesis extend the results of J.J.andnbsp;Seidel on equiangular lines.</p> <p>Regular 2-graphs are constructed which admit the PSL(2,q) , qandequiv;l (mod 4), Sp(2m,2), in both doubly transitive representations; PSU(3,q<sup>2</sup>), q odd; all groups of Ree type together with <sup>2</sup>G<sub>2</sub>(3) = Aut(PSL(2,8)); the sporadic simple groups C<sub>3</sub> and HiS; the group V.Sp(2m,2) which is the semi-direct product of the group V of translations of a vector space of dimension 2m over the field GF(2) by Sp(2m,2). By studying the centraliser ring of a monomial representation associated with the doubly transitive representation it is shown that (with the possible exception of some groups with a regular normal subgroup) the above groups are the only known groups which can act as doubly transitive groups of automorphisms of a regular 2-graph.</p> |
| first_indexed | 2024-03-07T05:19:46Z |
| format | Thesis |
| id | oxford-uuid:de7a3365-724c-4924-ab65-271a561dca20 |
| institution | University of Oxford |
| language | English |
| last_indexed | 2024-03-07T05:19:46Z |
| publishDate | 1971 |
| record_format | dspace |
| spelling | oxford-uuid:de7a3365-724c-4924-ab65-271a561dca202022-03-27T09:32:30ZSome topics in the theory of finite groupsThesishttp://purl.org/coar/resource_type/c_db06uuid:de7a3365-724c-4924-ab65-271a561dca20EnglishPolonsky Theses Digitisation Project1971Taylor, DTaylor, Donald E.Higman, G<p>A regular 2-graph consists of a set andOmega; together with a (non-empty) set <em>t</em> of three-element subsets of andOmega; such that any two-element subset of andOmega; is contained in the same number of elements of <em>t</em>, any four-element subset of andOmega; contains an even number of elements of <em>t</em> and not every three-element subset of andOmega; is in <em>t</em>. These objects were introduced by G.andnbsp;Higman who used a regular 2-graph with 276 points to provide a combinatorial setting for the doubly transitive representation of Conway's sporadic simple group C<sub>3</sub>.</p> <p>In this thesis it is shown that regular 2-graphs are in one-one correspondence with equivalence classes of strong graphs (as defined by J.J.andnbsp;Seidel). Moreover, for each point of a regular 2-graph there is a natural way of defining a strongly regular graph on the remaining points. These graphical representations are used to obtain restrictions on the structure and on the parameters of a regular 2-graph. It is also possible, via the strong graphs, to represent a regular 2-graph as a configuration of equiangular lines in Euclidean space. Conversely, results about regular 2-graphs obtained in this thesis extend the results of J.J.andnbsp;Seidel on equiangular lines.</p> <p>Regular 2-graphs are constructed which admit the PSL(2,q) , qandequiv;l (mod 4), Sp(2m,2), in both doubly transitive representations; PSU(3,q<sup>2</sup>), q odd; all groups of Ree type together with <sup>2</sup>G<sub>2</sub>(3) = Aut(PSL(2,8)); the sporadic simple groups C<sub>3</sub> and HiS; the group V.Sp(2m,2) which is the semi-direct product of the group V of translations of a vector space of dimension 2m over the field GF(2) by Sp(2m,2). By studying the centraliser ring of a monomial representation associated with the doubly transitive representation it is shown that (with the possible exception of some groups with a regular normal subgroup) the above groups are the only known groups which can act as doubly transitive groups of automorphisms of a regular 2-graph.</p> |
| spellingShingle | Taylor, D Taylor, Donald E. Some topics in the theory of finite groups |
| title | Some topics in the theory of finite groups |
| title_full | Some topics in the theory of finite groups |
| title_fullStr | Some topics in the theory of finite groups |
| title_full_unstemmed | Some topics in the theory of finite groups |
| title_short | Some topics in the theory of finite groups |
| title_sort | some topics in the theory of finite groups |
| work_keys_str_mv | AT taylord sometopicsinthetheoryoffinitegroups AT taylordonalde sometopicsinthetheoryoffinitegroups |