Bounds on the number of lifts of a Brauer character in a p-solvable group

The Fong-Swan theorem shows that for a $p$-solvable group $G$ and Brauer character \phi \in \ibrg, there is an ordinary character \chi \in \irrg such that $\chi^0 = \phi$, where $^0$ denotes restriction to the $p$-regular elements of $G$. This still holds in the generality of $\pi$-separable groups \cite{bpi}, where \ibrg is replaced by \ipig. For \phi \in \ipig, let L_{\phi} = \{\chi \in \irrg \mid \chi^0 = \phi \}. In this paper we give a lower bound for the size of $L_{\phi}$ in terms of the structure of the normal nucleus of $\phi$ and, if $G$ is assumed to be odd and $\pi = \{p' \}$, we give an upper bound for $L_{\phi}$ in terms of the vertex subgroup for $\phi$

Lifts and vertex pairs in solvable groups

Suppose $G$ is a $p$-solvable group, where $p$ is odd. We explore the connection between lifts of Brauer characters of $G$ and certain local objects in $G$, called vertex pairs. We show that if $\chi$ is a lift, then the vertex pairs of $\chi$ form a single conjugacy class. We use this to prove a sufficient condition for a given pair to be a vertex pair of a lift and to study the behavior of lifts with respect to normal subgroups

Counting lifts of Brauer characters

In this paper we examine the behavior of lifts of Brauer characters in p-solvable groups where p is an odd prime. In the main result, we show that if \phi \in IBrp(G) is a Brauer character of a solvable group such that \phi has an abelian vertex subgroup Q, then the number of lifts of \phi in Irr(G) is at most |Q|. In order to accomplish this, we develop several results about lifts of Brauer characters in p-solvable groups that were previously only known to be true in the case of groups of odd order.Comment: A different proof of Theorem 1 is in the paper "The number of lifts of Brauer characters with a normal vertex" by J.P. Cossey, M.L.Lewis, and G. Navarro. Hence, we do not expect to try to publish this note. We feel that the proof in this paper is of independent interes

Controlling composition factors of a finite group by its character degree ratio

For a finite nonabelian group $G$ let \rat(G) be the largest ratio of degrees of two nonlinear irreducible characters of $G$. We show that nonabelian composition factors of $G$ are controlled by \rat(G) in some sense. Specifically, if $S$ different from the simple linear groups \PSL_2(q) is a nonabelian composition factor of $G$, then the order of $S$ and the number of composition factors of $G$ isomorphic to $S$ are both bounded in terms of \rat(G). Furthermore, when the groups \PSL_2(q) are not composition factors of $G$, we prove that |G:\Oinfty(G)|\leq \rat(G)^{21} where \Oinfty(G) denotes the solvable radical of $G$.Comment: 16 pages, 1 tabl

On a conjecture of Gluck

Let $F(G)$ and $b(G)$ respectively denote the Fitting subgroup and the largest degree of an irreducible complex character of a finite group $G$. A well-known conjecture of D. Gluck claims that if $G$ is solvable then $|G:F(G)|\leq b(G)^{2}$. We confirm this conjecture in the case where $|F(G)|$ is coprime to 6. We also extend the problem to arbitrary finite groups and prove several results showing that the largest irreducible character degree of a finite group strongly controls the group structure.Comment: 16 page

Research Statement

My primary area of research is the representation theory of finite groups. The main focus of my work is the representation theory of solvable groups, and especially the interplay between the representations of a group over a field of characteristic zero and the representations over a field with positive characteristic. In addition, I am interested in th