Neumann, P.M. (1966) A study of some finite permutation groups. PhD thesis, University of Oxford.

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Abstract
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 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 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.
Item Type:  Thesis (PhD) 

Subjects:  D  G > Group theory and generalizations 
Research Groups:  Algebra Research Group 
ID Code:  888 
Deposited By:  Eprints Administrator 
Deposited On:  15 Jan 2010 12:14 
Last Modified:  29 May 2015 18:34 
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