Title:

A study of some finite permutation groups

This thesis records an attempt to prove the two conjecture: Conjecture A: Every finite nonregular primitive permutation group of degree n contains permutations fixing one point but fixing at most $n^{1/2}$ points. Conjecture C: Every finite irreducible linear group of degree m > 1 contains an element whose fixedpoint space has dimension at most m/2. Variants of these conjectures are formulated, and C is reduced to a special case of A. The main results of the investigation are: Theorem 2: Every finite nonregular primitive permutation group of degree n contains permutations which fix one point but fix fewer than (n+3)/4 points. Theorem 3: Every finite nonregular primitive soluble permutation group of degree n contains permutations which fix one point but fix fewer than $n^{7/18}$ points. Theorem 4: If H is a finite group, F is a field whose characteristic is 0 or does not divide the order of H, and M is a nontrivial irreducible Hmodule of dimension m over F, then there is an element h in H whose fixedpoint space in M has dimension less than m/2. Theorem 5: If H is a finite soluble group, F is any field, and M is a nontrivial irreducible Hmodule of dimension m over F, then there is an element h in H whose fixedpoint space in M has dimension less than 7m/18. Proofs of these assertions are to be found in Chapter II; examples which show the limitations on possible strenghtenings of the conjectures and results are marshalled in Chapter III. A detailed formulation of the problems and results is contained in section 1.
