Abstract
Suppose V is a completely reducible faithful G-module for a 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 in some ways best possible. This is applied to generate many theorems showing that a solvable group must have characters of large degree.
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