Finite Permutation Groups

The major part of my thesis is concerned with the size and structure of Sylow p-subgroups 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 Ω of degree n, and if P is a Sylow p-subgroup of G for some prime p dividing |G|, then the number of points of Ω fixed by P is less than n⁄2.
Theorem 2.2. Let G be a primitive permutation group on Ω of degree n = kp, where p is a prime, and such that G does not contain the alternating group An. Let P be a Sylow p-subgroup of G, and suppose that P has no orbits of length greater thin p. Then P has order p unless
|P| = 4 and G is PSL(2,5) permuting the 6 points or the 1-dimensional projective geometry PG(1,5), or
|P| = 9 and G is the Mathieu group M11 in its 3-transitive representation of degree 12.
This result is due to L. Scott for the case in which G is not 2-transitive and my contribution is the 2-transitive case.
Theorem 2.3. Let G be a 2-transitive permutation group on Ω of degree n = kp + f, for some prime p, such that G does not contain the alternating group An. Suppose that p divides |G| and that a Sylow p-subgroup P of G has k orbits of length p and f fixed points in Ω. Then P has order p unless f = 0.
As the first application of these results we prove Theorem 7.1 below about 2-transitive groups of degree r2 + 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 2-transitive permutation group on Ω of degree n = r2 + 3r + 3, where r is a prime greater than 3, and such that r divides |G|, then either G contains the alternating group An, or r is of the form 2m - 1, a Mersenne prime, for some odd prime m, and G is such that PSL(3,2m) ≤ G ≤ PΓL(3,2m).
Next we turn to 2-transitive groups of degree p2, where p is a prime. In looking at the case whore the Sylow p-subgroups are cyclic, the situation arose in which G had an indecomposable representation of degree less than |P|⁄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 p-subgroup P of order pk ≥ p2, which is a T.I. set. Suppose that G is not p-soluble. Suppose that G has an indecomposable representation ℒ in a field K of characteristic p of degree d ≤ pk, such that P is not contained in the kernel of ℒ. Then ℒp is indecomposable, CG(P) = PxZ(G), and d ≥ (pk+1)⁄2.
Finally there are some results about 2-transitive groups of degree p2, following on from Wielendt's classification of the simply transitive groups:
Theorem 12.3. If G is a 2-transitive group of degree p2 and P is a Sylow p-subgroup of G, then either
|P| ≥ p4 and G contains Ap2, for p ≥ 3, or
|P| = p3 and G ≤ Aff(2,p), (and G has PSL(2,p) as a composition factor), or
|P| = 33 and G is PΓL(2,8) of degree 9, or
|P| = 23 and G is S4 of degree 4, or
|P| = p2.
If G is primitive of degree pk and its Sylow p-subgroups are cyclic, we use Theorem 9.2 to extend results of Neumann and Ito, (Theorem 14.2, and Corollary 14.3).