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Title: Rank 3 permutation groups with a regular normal subgroup
Author: Hill, Raymond
Awarding Body: University of Warwick
Current Institution: University of Warwick
Date of Award: 1971
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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 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, 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.
Supervisor: Not available Sponsor: Science Research Council
Qualification Name: Thesis (Ph.D.) Qualification Level: Doctoral
EThOS ID:  DOI: Not available
Keywords: QA Mathematics