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Chris Parker

Publications and source records attributed to Chris Parker.

At least 19 recordsLinked to original sources
arXiv2026

Modules with few Jordan blocks for rank $1$ groups of Lie type and related groups

Suppose that $p$ is a prime and $X$ is a finite group with a strongly $p$-embedded subgroup, for example a rank $1$ group of Lie type in characteristic $p$. Let $m$ denote the $p$-rank of $X$ and assume that $m \ge 2$. We say that a faithful $\FF_p X$-module is $k$-active if some element of order $p$ in $X$ acts with exactly $k$ non-trivial Jordan blocks. In this paper, we determine the non-trivial composition factors of the faithful $\FF_pX$-modules which are $k$-active for some $k\leq m$.

math.GR↗
arXiv2026

The structure of finite groups invariably generated by two elements of prime order

A group $G$ is invariably generated by two elements $a$ and $b$ if $G=\langle a^g,b^h\rangle $ for all $g,h\in G$. This paper provides a structural description of finite groups invariably generated by two elements of distinct prime orders $s$ and $t$. We illustrate the use of our main result in a case study with $s=2$ and $t=3$. In particular we prove that an almost simple group is invariably generated by an element of order $2$ and an element of order $3$ if and only if it is isomorphic to $\mathrm{PGL}_2(3^{2^b})$ for some $b \ge 1$.

math.GR↗
arXiv2026

The Gill-Guillot commuting graph for sporadic and related groups

Let $G$ be a finite group and $\mathcal{C}$ a normal subset of $G$. The Gill-Guillot graph has vertex set $\mathcal C$ with distinct $x, y \in \mathcal C$ adjacent if and only if $x$ and $y$ commute and $\{xy^{-1},x^{-1}y\} \cap \mathcal C$ is non-empty. We study the connectivity of this graph for quasisimple groups with $G/Z(G)$ a sporadic simple group and for certain simple groups with exceptional Schur multiplier.

math.GR↗
arXiv2025

Groups with conjugacy classes of coprime sizes

Suppose that $x$, $y$ are elements of a finite group $G$ lying in conjugacy classes of coprime sizes. We prove that $\langle x^G \rangle \cap \langle y^G \rangle$ is an abelian normal subgroup of $G$ and, as a consequence, that if $x$ and $y$ are $π$-regular elements for some set of primes $π$, then $x^G y^G$ is a $π$-regular conjugacy class in $G$. The latter statement was previously known for $π$-separable groups $G$ and this generalisation permits us to extend several results concerning the common divisor graph on $p$-regular conjugacy classes, for some prime $p$.

math.GR↗
arXiv2025

Expansion of normal subsets of odd-order elements in finite groups

Let $G$ be a finite group and $K$ a normal subset consisting of odd-order elements. The rational closure of $K$, denoted $\mathbf D_K$, is the set of elements $x \in G$ with the property that $\langle x \rangle = \langle y \rangle$ for some $y$ in $K$. If $K^2 \subseteq \mathbf D_K$, we prove that $\langle K \rangle$ is soluble.

math.GR↗
arXiv2025

Generalizing imaging biomarker repeatability studies using Bayesian inference: Applications in detecting heterogeneous treatment response in whole-body diffusion-weighted MRI of metastatic prostate cancer

The assessment of imaging biomarkers is critical for advancing precision medicine and improving disease characterization. Despite the availability of methods to derive disease heterogeneity metrics in imaging studies, a robust framework for evaluating measurement uncertainty remains underdeveloped. To address this gap, we propose a novel Bayesian framework to assess the precision of disease heterogeneity measures in biomarker studies. Our approach extends traditional methods for evaluating biomarker precision by providing greater flexibility in statistical assumptions and enabling the analysis of biomarkers beyond univariate or multivariate normally-distributed variables. Using Hamiltonian Monte Carlo sampling, the framework supports both, for example, normally-distributed and Dirichlet-Multinomial distributed variables, enabling the derivation of posterior distributions for biomarker parameters under diverse model assumptions. Designed to be broadly applicable across various imaging modalities and biomarker types, the framework builds a foundation for generalizing reproducible and objective biomarker evaluation. To demonstrate utility, we apply the framework to whole-body diffusion-weighted MRI (WBDWI) to assess heterogeneous therapeutic responses in metastatic bone disease. Specifically, we analyze data from two patient studies investigating treatments for metastatic castrate-resistant prostate cancer (mCRPC). Our results reveal an approximately 70% response rate among individual tumors across both studies, objectively characterizing differential responses to systemic therapies and validating the clinical relevance of the proposed methodology. This Bayesian framework provides a powerful tool for advancing biomarker research across diverse imaging-based studies while offering valuable insights into specific clinical applications, such as mCRPC treatment response.

stat.AP↗
arXiv2025

Fusion systems related to polynomial representations of $\mathrm{SL}_2(q)$

Let $q$ be a power of a fixed prime $p$. We classify up to isomorphism all simple saturated fusion systems on a certain class of $p$-groups constructed from the polynomial representations of $\mathrm{SL}_2(q)$, which includes the Sylow $p$-subgroups of $\mathrm{GL}_3(q)$ and $\mathrm{Sp}_4(q)$ as special cases. The resulting list includes all Clelland--Parker fusion systems, a simple exotic fusion system discovered by Henke--Shpectorov, and a new infinite family of exotic examples.

math.GR↗
arXiv2025

The connectivity of the normalising and permuting graph of a finite soluble group

We introduce the normalising graph of a group and study the connectivity of the normalising and permuting graphs of a group when the group is finite and soluble. In particular, we classify finite soluble groups with disconnected normalising graph. The main results shows that if a finite soluble group has connected normalising graph then this graph has diameter at most 6. Furthermore, this bound is tight. A corollary then presents the connectivity properties of the permuting graph.

math.GR↗
arXiv2024

Lifting polynomial representations of $\mathrm{SL}_2(p^r)$ from $\mathbb{F}_p$ to $\mathbb{Z}/p^s\mathbb{Z}$

We describe all of the irreducible polynomial $\mathbb{F}_p\mathrm{SL}_2(p^r)$ representations which lift to $(\mathbb{Z}/p^s\mathbb{Z})\mathrm{SL}_2(p^r)$ representations for $s>1$, observing that they almost never do. We also show that two related indecomposable $\mathbb{F}_p \mathrm{SL}_2(p^r)$ representations cannot be lifted to $\mathbb{Z}/p^s\mathbb{Z}$ representations for $s>1$.

math.RT↗
arXiv2023

Vertex stabilizers of locally $s$-arc transitive graphs of pushing up type

Suppose that $Δ$ a thick, locally finite and locally $s$-arc transitive $G$-graph with $s \ge 4$. For a vertex $z$ in $Δ$, let $G_z$ be the stabilizer of $z$ and $G_z^{[1]}$ be the kernel of the action of $G_z$ on the neighbours of $z$. We say $Δ$ is of pushing up type provided there exists a prime $p$ and a $1$-arc $(x,y)$ such that $C_{G_z}(O_p(G_z^{[1]})) \le O_p(G_z^{[1]})$ for $z \in \{x,y\}$ and $O_p(G_x^{[1]}) \le O_p(G_y^{[1]})$. We show that if $Δ$ is of pushing up type, then $O_p(G_x^{[1]})$ is elementary abelian and $G_x/G_x^{[1]}\cong X$ with ${\rm PSL}_2(p^a)\le X \le {\rm PΓL}_2(p^a)$.

math.GR↗
arXiv2023

Squares of conjugacy classes and a variant on the Baer-Suzuki Theorem

For $p$ a prime, $G$ a finite group and $A$ a normal subset of elements of order $p$, we prove that if $A^2 = \{ab \mid a, b \in A\}$ consists of $p$-elements then $Q = \langle A \rangle$ is soluble. Further, if $O_p(G) = 1$, we show that $p$ is odd, $F(Q)$ is a non-trivial $p'$-group and $Q/F(Q)$ is an elementary abelian $p$-group. We also provide examples which show this conclusion is best possible.

math.GR↗
arXiv2021

Algorithms for fusion systems with applications to $p$-groups of small order

For a prime $p$, we describe a protocol for handling a specific type of fusion system on a $p$-group by computer. These fusion systems contain all saturated fusion systems. This framework allows us to computationally determine whether or not two subgroups are conjugate in the fusion system for example. We describe a generation procedure for automizers of every subgroup of the $p$-group. This allows a computational check of saturation. These procedures have been implemented using MAGMA. We describe a program to search for saturated fusion systems $\mathcal{F}$ on $p$-groups with $O_p(\mathcal{F})=1$ and $O^p(\mathcal{F})=\mathcal{F}$. Employing these computational methods we determine all such fusion system on groups of order $p^n$ where $(p,n) \in \{(3,4),(3,5),(3,6),(3,7),(5,4),(5,5),(5,6),(7,4),(7,5)\}$. This gives the first complete picture of which groups can support saturated fusion systems on small $p$-groups of odd order.

math.GR↗
arXiv2019

The local structure theorem, the non-characteristic 2 case

Let $p$ be a prime, $G$ a finite $\mathcal{K}_p$-group, $S$ a Sylow $p$-subgroup of $G$ and $Q$ be a large subgroup of $G$ in $S$. The aim of the Local Structure Theorem is to provide structural information about subgroups $L$ with $S \leq L$, $O_p(L) \not= 1$ and $L \not\leq N_G(Q)$. There is, however, one configuration where no structural information about $L$ can be given using the methods in the proof of the Local Structure Theorem. In this paper we show that for $p=2$ this hypothetical configuration cannot occur. We anticipate that our theorem will be used in the programme to revise the classification of the finite simple groups.

math.GR↗
arXiv2019

The Local Structure Theorem: The wreath product case

Groups with a large $p$-subgroup, $p$ a prime, include almost all of the groups of Lie type in characteristic $p$ and so the study of such groups adds to our understanding of the finite simple groups. In this article we study a special class of such groups which appear as wreath product cases of the Local Structure Theorem.

math.GR↗
arXiv2017

Fusion systems over a Sylow $p$-subgroup of $\mathrm{G}_2(p)$

For $S$ a Sylow $p$-subgroup of the group $\mathrm{G}_2(p)$ for $p$ odd, up to isomorphism of fusion systems, we determine all saturated fusion systems $\mathcal{F}$ on $S$ with $O_p(\mathcal{F})=1$. For $p \ne 7$, all such fusion systems are realized by finite groups whereas for $p=7$ there are $29$ saturated fusion systems of which $27$ are exotic.

math.GR↗