| Suppose V is a completely reducible faithful G-module for a finite solvable group G, we show G has a "large" orbit on V. Specifically, there is v ∈ V such that C G(v) is contained in a normal subgroup of derived length 9 contained in the seventh ascending Fitting subgroup of G. This is applied to generate many theorems showing that a solvable group must have characters of large degree.;Suppose that a finite solvable group G acts faithfully, irreducibly and quasi-primitively on a finite vector space V. Then G has a uniquely determined normal subgroup E which is a direct product of extraspecial p-groups for various p and we denote e = &vbm0;E/ZE &vbm0; . We prove that when e ≥ 10 and e ≠ 16, G will have at least 5 regular orbits on V. We also construct groups with no regular orbits on V when e = 8, 9 and 16. (Full text of this dissertation may be available via the University of Florida Libraries web site. Please check http://www.uflib.ufl.edu/etd.html)... |
