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Shawn T. Burkett

Publications and source records attributed to Shawn T. Burkett.

12 recordsLinked to original sources
arXiv2024

A Frobenius group analog for Camina triples

Frobenius groups are an object of fundamental importance in finite group theory. As such, several generalizations of these groups have been considered. Some examples include: A Frobenius--Wielandt group is a triple $(G,H,L)$ where $H/L$ is {\it almost} a Frobenius complement for $G$; A Camina pair is a pair $(G,N)$ where $N$ is {\it almost} a Frobenius kernel for $G$; A Camina triple is a triple $(G,N,M)$ where $(G,N)$ and $(G,M)$ are {\it almost} Camina pairs. In this paper we study triples $(G,N,M)$ where $(G,N)$ and $(G,M)$ are {\it almost} Frobenius groups.

math.GR↗
arXiv2021

Partial GVZ-groups

Following the literature, a group $G$ is called a group of central type if $G$ has an irreducible character that vanishes on $G\setminus Z(G)$. Motivated by this definition, we say that a character $χ\in {\rm Irr}(G)$ has central type if $χ$ vanishes on $G\setminus Z(χ)$, where $Z(χ)$ is the center of $χ$. Groups where every irreducible character has central type have been studied previously under the name GVZ-groups (and several other names) in the literature. In this paper, we study the groups $G$ that possess a nontrivial, normal subgroup $N$ such that every character of $G$ either contains $N$ in its kernel or has central type. The structure of these groups is surprisingly limited and has many aspects in common with both central type groups and GVZ-groups.

math.GR↗
arXiv2020

GVZ-groups, Flat Groups, and CM-Groups

We show that a group is a GVZ-group if and only if it is a flat group. We show that the nilpotence class of a GVZ-group is bounded by the number of distinct degrees of irreducible characters. We also show that certain CM-groups can be characterized as GVZ-groups whose irreducible character values lie in the prime field.

math.GR↗
arXiv2020

Toward a Classification of the Supercharacter Theories of $C_p\times C_p$

In this paper, we study the superscharacter theories of elementary abelian $p$-groups of order $p^2$. We show that the supercharacter theories that arise from the direct product construction and the $\ast$-product construction can be obtained from automorphisms. We also prove that any supercharacter theory of an elementary abelian $p$-group of order $p^2$ that has a nonidentity superclass of size $1$ or a nonprincipal linear supercharacter must come from either a $\ast$-product or a direct product. Although we are unable to prove results for general primes, we do compute all of the supercharacter theories when $p = 2, 3, 5$, and based on these computations along with particular computations for larger primes, we make several conjectures for a general prime $p$.

math.GR↗
arXiv2020

Supercharacter theory via the group determinant

Ferdinand Georg Frobenius is generally considered the creator of character theory of finite groups. This achievement came from the study of the group determinant, which is the determinant of a matrix coming from the regular representation. In this paper, we generalize several of Frobenius' results about the group determinant and use them find a new formulation of supercharacter theory in terms of factorizations of the group determinant.

math.RT↗
arXiv2019

Characterizations of nested GVZ-groups by central series

Many properties of groups can be defined by the existence of a particular normal series. The classic examples being solvability, supersolvability and nilpotence. Among the nilpotent groups are the so-called nested GVZ-groups --- groups where the centers of the irreducible characters form a chain, and where every irreducible character vanishes off of its center. In this paper, we show that nested GVZ-groups can be characterized by the existence of a certain ascending central series, or by the existence of a certain descending central series.

math.GR↗
arXiv2019

Vanishing-off subgroups and supercharacter theory products

In this paper, we study the vanishing-off subgroups of supercharacters, and use these to determine several new characterizations of supercharacter theory products. In particular, we give a character theoretic characterization that allows us to conclude that one may determine if a supercharacter theory is a $Δ$-product or $\ast$-product from the values of its corresponding supercharacters.

math.RT↗
arXiv2019

A Jördan-Holder type theorem for supercharacter theories

The Jordan-Hölder Theorem is a general term given to a collection of theorems about maximal chains in suitably nice lattices. For example, the well-known Jordan-Hölder type theorem for chief series of finite groups has been rather useful in studying the structure of finite groups. In this paper, we present a Jordan-Hölder type theorem for supercharacter theories of finite groups, which generalizes the one for chief series of finite groups.

math.GR↗
arXiv2018

An analog of nilpotence arising from supercharacter theory

The goal of this paper is to generalize several group theoretic concepts such as the center and commutator subgroup, central series, and ultimately nilpotence to a supercharacter theoretic setting, and to use these concepts to show that there can be a strong connection between the structure of a group and the structure of its supercharacter theories. We then use these concepts to show that the upper and lower annihilator series of $J$ can be described in terms of certain central series for the algebra group $G=1+J$ defined by $\mathsf{S}$, when $\mathsf{S}$ is the algebra group supercharacter theory defined by Diaconis--Isaacs.

math.GR↗