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Rank 3 permutation groups with a regular normal subgroup
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Hill, Raymond (1971) Rank 3 permutation groups with a regular normal subgroup. PhD thesis, University of Warwick.
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Official URL: http://webcat.warwick.ac.uk/record=b1733327~S1
Abstract
A (p ,n) group G is a permutation group (on a set Ω) which possesses a regular normal elementary abelian subgroup of order pn. The set Ω may be identified with a vector space V on which Go, the stabilizer of a point in G, acts as a subgroup of the general linear group GL(n,p). By a line of a subset ∆ of V, we mean the intersection of ∆ with a one-dimensional subspace of V. The main result (Theorem 1.3.2) concerns (*) - groups, the term we give to rank 3 (p,n; groups in which the stabilizer of a point is doubly-transitive on the lines of a suborbit. The essence or the problem is that of finding those subgroups of PGL (n,p) which have two orbits on the projective space PG (n – 1,p) and act doubly-transitively on one of them.
The notion of rank of a permutation group is discussed in 1.1, outline D.G. Higman’s combinatorial treatment of rank 3 groups.
Associated with each permutation group having a regular subgroup is a certain S - ring, an algebraic structure which is basic to our theory. In 2.1 we define parameters of a rank 3 S - ring whd.ch coincide with those of any associated rank :3 group. Hence (*) - group with given parameters may be classified by finding all S - rings with the same parameters and then finding the associated (*) - groups. To assist in this task the concepts of residual S-ring and the automorphism group of an S-ring are introduced. Also of great value is Tamaschke’s notion of' the dual S-ring, whi.ch is adapted to use in 2.2.
In 3.1 we see how the imposition of conditions of transitivity on a suborbit of a rank 3 (p,n) groups leads to information about the parameters. In 3.3 the various relations connecting the parameters of' a (*)- group are combined to yield specific sets of parameters, all of which are found in §4: to admit rank 3 S - rings. From results concerning the uniqueness of these S – rings, certain finite simple groups are characterised as their automorphism groups, and the proof of the main theorem is completed. A number of results are obtained as by – products in §4:, notably the answer to a question raised by Wielandt and a new representation of the simple group PSL(3,4) as a subgroup of PO-(6,3, leading to an interesting presentation of a recently-discovered balanced block design.
§5 is devoted to rank 3 (p,n) groups in which the transitivity condition on Go is replaced by the condition that the associated block design is balanced.
Item Type: | Thesis (PhD) | ||||
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Subjects: | Q Science > QA Mathematics | ||||
Library of Congress Subject Headings (LCSH): | Permutation groups, Group theory, Abelian groups | ||||
Official Date: | 1971 | ||||
Dates: |
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Institution: | University of Warwick | ||||
Theses Department: | Mathematics Institute | ||||
Thesis Type: | PhD | ||||
Publication Status: | Unpublished | ||||
Supervisor(s)/Advisor: | Green, J. A. (James Alexander) | ||||
Sponsors: | Science Research Council (Great Britain) | ||||
Extent: | 109 leaves | ||||
Language: | eng |
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