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Title:  Finite permutation groups : on the Sylow subgroups of primitive permutation groups  
Author:  Praeger, Cheryl E.  
Awarding Body:  University of Oxford  
Current Institution:  University of Oxford  
Date of Award:  1973  
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Abstract:  
The major part of my thesis is concerned with the size and structure of Sylow psubgroups of a primitive permutation group. The results of Theorems 2.2 and 2.3 were suggested by similar results of Jordan, Manning, Waiss, and othera, about elements of order p in a primitive group. The following are the three main results: Theorem 2.1. If G is a transitive permutation group on a set Î© of degree n, and if P is a Sylow psubgroup of G for some prime p dividing G, then the number of points of Î© fixed by P is less than ^{n}â_{2}. Theorem 2.2. Let G be a primitive permutation group on Î© of degree n = kp, where p is a prime, and such that G does not contain the alternating group A_{n}. Let P be a Sylow psubgroup of G, and suppose that P has no orbits of length greater thin p. Then P has order p unless
This result is due to L. Scott for the case in which G is not 2transitive and my contribution is the 2transitive case. Theorem 2.3. Let G be a 2transitive permutation group on Î© of degree n = kp + f, for some prime p, such that G does not contain the alternating group A_{n}. Suppose that p divides G and that a Sylow psubgroup P of G has k orbits of length p and f fixed points in Î©. Then P has order p unless f = 0. As the first application of these results we prove Theorem 7.1 below about 2transitive groups of degree r^{2} + 3r + 3, where r is a prime. This problem arose from a conjecture about transitive groups of prime degree, and work of Peter Neumann and Tom McDonough. Theorem 7.1. If G is a 2transitive permutation group on Î© of degree n = r^{2} + 3r + 3, where r is a prime greater than 3, and such that r divides G, then either G contains the alternating group A_{n}, or r is of the form 2^{m}  1, a Mersenne prime, for some odd prime m, and G is such that PSL(3,2^{m}) â¤ G â¤ PÎL(3,2^{m}). Next we turn to 2transitive groups of degree p^{2}, where p is a prime. In looking at the case whore the Sylow psubgroups are cyclic, the situation arose in which G had an indecomposable representation of degree less than ^{P}â_{2}. To deal with this, the next theorem, an extension of a result of Felt, was proved. Theorem 9.2. Let G be a finite group with a cyclic Sylow psubgroup P of order p^{k} â¥ p^{2}, which is a T.I. set. Suppose that G is not psoluble. Suppose that G has an indecomposable representation â in a field K of characteristic p of degree d â¤ p^{k}, such that P is not contained in the kernel of â. Then â_{p} is indecomposable, C_{G}(P) = PxZ(G), and d â¥ ^{(pk+1)}â_{2}. Finally there are some results about 2transitive groups of degree p^{2}, following on from Wielendt's classification of the simply transitive groups: Theorem 12.3. If G is a 2transitive group of degree p^{2} and P is a Sylow psubgroup of G, then either
If G is primitive of degree p^{k} and its Sylow psubgroups are cyclic, we use Theorem 9.2 to extend results of Neumann and Ito, (Theorem 14.2, and Corollary 14.3). 

Supervisor:  Not available  Sponsor:  Not available  
Qualification Name:  Thesis (Ph.D.)  Qualification Level:  Doctoral  
EThOS ID:  uk.bl.ethos.285414  DOI:  Not available  
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