arXiv subjects

James P. Cossey

Publications and source records attributed to James P. Cossey.

10 recordsLinked to original sources

arXiv2026

A height-zero type result for blocks of solvable groups

Let $B$ be a $p$-block of a finite group $G$ with defect group $D$. The more difficult direction of the recently proven height zero conjecture says that $D$ is abelian if every character in Irr$(B)$ has height zero. We consider a smaller set than Irr$(B)$. In particular, if $φ\in {\rm IBr}_p(B)$, we let Irr$(φ)$ be the set of characters $χ\in {\rm Irr}(G)$ such that $φ$ is a constituent of $χ^o$. Now suppose $G$ is solvable and $φ$ is a height zero Brauer character in some block $B$ of $G$ with defect group $D$. Here we show that if every character in Irr$(φ)$ has height zero, then the defect group $D$ of the block containing $φ$ is abelian for $p \geq 5$ and almost abelian for $p = 2$ or $3$. This has a nice consequence for primitive characters of $p$-complements in solvable groups.

math.GR↗

arXiv2025

Sylow subgroups and the number of irreducible characters of degrees divisible by a prime $p$

Let $G$ be a finite group and $p$ a prime. We establish an upper bound for the derived length of a Sylow $p$-subgroup of $G$ in terms of the number of irreducible characters of $G$ whose degrees are divisible by $p$. We also prove that if $B$ is a $p$-block of a finite $p$-solvable group $G$ with defect group $D$, then the derived length of $D$ is at most one more than the number of ordinary irreducible characters of positive height in $B$.

math.GR↗

arXiv2014

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.

math.GR↗

arXiv2013

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 $G$.

math.GR↗

arXiv2011

Counting characters in blocks of solvable groups with abelian defect group

If $G$ is a solvable group and $p$ is a prime, then the Fong-Swan theorem shows that given any irreducible Brauer character $ϕ$ of $G$, there exists a character $χ\in \irrg$ such that $χ^o = ϕ$, where $^o$ denotes the restriction of $χ$ to the $p$-regular elements of $G$. We say that $χ$ is a {\it{lift}} of $ϕ$ in this case. It is known that if $ϕ$ is in a block with abelian defect group $D$, then the number of lifts of $ϕ$ is bounded above by $|D|$. In this paper we give a necessary and sufficient condition for this bound to be achieved, in terms of local information in a subgroup $V$ determined by the block $B$. We also apply these methods to examine the situation when equality occurs in the $k(B)$ conjecture for blocks of solvable groups with abelian defect group.

math.GR↗

arXiv2010

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 $χ$ is a lift, then the vertex pairs of $χ$ 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.

math.GR↗

arXiv2010

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 ϕ\in IBrp(G) is a Brauer character of a solvable group such that ϕhas an abelian vertex subgroup Q, then the number of lifts of ϕin 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.

math.GR↗

arXiv2006

A construction of two distinct canonical sets of lifts of Brauer characters of a p-solvable group

Navarro defined the set ${Irr}(G \mid Q, δ) \subseteq {Irr}(G)$, where $Q$ is a $p$-subgroup of a $p$-solvable group $G$, and shows that if $δ$ is the trivial character of $Q$, then ${Irr}(G \mid Q, δ)$ provides a set of canonical lifts of ${\textup{IBr}}_p(G)$, the irreducible Brauer characters with vertex $Q$. Previously, Isaacs defined a canonical set of lifts $\bpig$ of $\ipig$. Both of these results extend the Fong-Swan Theorem to $π$-separable groups, and both construct canonical sets of lifts of the generalized Brauer characters. It is known that in the case that $2 \in π$, or if $|G| $ is odd, we have $\bpig = {Irr}(G \mid Q, 1_Q)$. In this note we give a counterexample to show that this is not the case when $2 \not\in π$. It is known that if $N \nrml G$ and $χ\in \bpig$, then the constituents of $χ_N$ are in $\bpi(N)$. However, we use the same counterexample to show that if $N \nrml G$, and $χ\in {Irr}(G\mid Q, 1_Q)$ is such that $θ\in {Irr}(N)$ and $[θ, χ_N] \neq 0$, then it is not necessarily the case that $θ\in \textup{Irr}(N)$ inherits this property.

math.GR↗

arXiv2006

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

math.GR↗

arXiv2006

Constructing all irreducible Specht modules in a block of the symmetric group

For any prime p, we construct, and simultaneously count, all of the complex Specht modules in a given p-block of the symmetric group which remain irreducible when reduced modulo p. We call the Specht modules with this property p-irreducible modules. Recently Fayers has proven a conjecture of James and Mathas that provides a characterization of the partitions that correspond to the p-irreducible modules. In this paper we present a method for decomposing the partitions corresponding to p-irreducible modules, and we use this decomposition to construct and count all of the partitions corresponding to p-irreducible Specht modules in a given block.

math.CO↗