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ABSTRACT. XIE, NANDI. Ph.D. March 1996.

Mathematics, Ohio University.

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Director of Dissertation: Thomas R. Wolf.

The main result of this dissertation is a logarithmic bound for the
derived length of a primitive solvable permutation
group in terms of its rank, the number of orbits of its point
stabilizer.
As an application of this result, a logarithmic bound for the derived
length of the quotient group G/F(G) of a solvable group G by its Fitting
subgroup F(G) is obtained in terms of the number of irreducible characters
of G.

Also, a logarithmic bound for the derived length of a solvable
group A acting on a solvable group G with coprime order
is found in terms of the number of orbits of A acting on the
set of irreducible characters of G, as well as the number of
orbits of A acting on the set of irreducible Brauer characters of G.

An attempt is made to bound the derived length of the linear group G of
a solvable permutation group in terms of the number of orbit sizes
of G; partial results are obtained. The derived length of the linear
group G of a solvable permutation group S with one nontrivial G-orbit
size (i.e., S is a 3/2-transitive permutation group) is less than or
equal to 6.

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