Master of Science in Mathematics, Youngstown State University, 2011, Department of Mathematics and Statistics

In 1961 Roger W. Carter proved a theorem about solvable groups similar to Sylow's theorem. He proved that if a group is solvable then it always contains a nilpotent, self-normalizing subgroup called a Carter subgroup, and that all such subgroups are conjugate to each other by an element of the group. The objective of this thesis is to present a proof of Carter's theorem.

Committee: Neil Flowers PhD (Advisor); Thomas Wakefield PhD (Committee Member); Eric Wingler PhD (Committee Member) Subjects: Mathematics

PHD, Kent State University, 2021, College of Arts and Sciences / Department of Mathematical Sciences

We are interested in determining the bound of the average of the degrees of the irreducible characters whose degrees are not divisible by some prime $p$ that guarantees a finite group $G$ of odd order is $p$-nilpotent. We find a bound that depends on the prime $p$. If we further restrict our average by fixing a subfield $k$ of the complex numbers and then compute the average of the degrees of the irreducible characters whose degrees are not divisible by $p$ and have values in $k$, then we will see that we obtain a bound that depends on both $p$ and $k$. Moreover, we find examples that make those bounds best possible

Committee: L. Mark Lewis Prof. (Advisor); M. Gagola Stephen Prof. (Committee Member); L. White Donald Prof. (Committee Member) Subjects: Mathematics