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 $n^{1/2}$ 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 $n^{7/18}$ 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 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.
| Identifer | oai:union.ndltd.org:bl.uk/oai:ethos.bl.uk:580729 |
| Date | January 1966 |
| Creators | Neumann, Peter M. |
| Publisher | University of Oxford |
| Source Sets | Ethos UK |
| Detected Language | English |
| Type | Electronic Thesis or Dissertation |
| Source | http://ora.ox.ac.uk/objects/uuid:13662173-b0cd-4776-bec4-e4f31eaa654b |
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