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 non-regular 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 fixed-point 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 non-regular primitive permutation group of degree n contains permutations which fix one point but fix fewer than (n+3)/4 points.

Theorem 3: Every finite non-regular 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 non-trivial irreducible H-module of dimension m over F, then there is an element h in H whose fixed-point 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 non-trivial irreducible H-module of dimension m over F, then there is an element h in H whose fixed-point 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) |
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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: | 15 Jan 2010 12:14 |

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