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S. P. Glasby

Publications and source records attributed to S. P. Glasby.

At least 37 records · Page 2Linked to original sources
arXiv2020

Conjugacy class sizes in arithmetic progression

Let ${\rm cs}(G)$ denote the set of conjugacy class sizes of a group $G$, and let ${\rm cs}^*(G)={\rm cs}(G)\setminus\{1\}$ be the sizes of non-central classes. We prove three results. We classify all finite groups $G$ with ${\rm cs}(G)=\{a, a+d, \dots ,a+rd\}$ an arithmetic progression with $r\geqslant 2$. (We show that ${\rm cs}(G)=\{1,2,3\}$.) Our most substantial result classifies all $G$ with ${\rm cs}^*(G)=\{2,4,6\}$. Finally, we classify all groups $G$ whose largest two non-central conjugacy class sizes are coprime. (Here it is not obvious but it is true that ${\rm cs}^*(G)$ has two elements, and so is an arithmetic progression.)

math.GR↗
arXiv2019

Involution centralisers in finite unitary groups of odd characteristic

We analyse the complexity of constructing involution centralisers in unitary groups over fields of odd order. In particular, we prove logarithmic bounds on the number of random elements required to generate a subgroup of the centraliser of a strong involution that contains the last term of its derived series. We use this to strengthen previous bounds on the complexity of recognition algorithms for unitary groups in odd characteristic. Our approach generalises and extends two previous papers by the second author and collaborators on strong involutions and regular semisimple elements of linear groups.

math.GR↗
arXiv2019

Classifying uniformly generated groups

A finite group $G$ is called *uniformly generated*, if whenever there is a (strictly ascending) chain of subgroups $1<\langle x_1\rangle<\langle x_1,x_2\rangle <\cdots<\langle x_1,x_2,\dots,x_d\rangle=G$, then $d$ is the minimal number of generators of $G$. Our main result classifies the uniformly generated groups without using the simple group classification. These groups are related to finite projective geometries by a result of Iwasawa on subgroup lattices.

math.GR↗
arXiv2019

Permutations with orders coprime to a given integer

Let $m$ be a positive integer and let $ρ(m,n)$ be the proportion of permutations of the symmetric group ${\rm Sym}(n)$ whose order is coprime to $m$. In 2002, Pouyanne proved that $ρ(n,m)n^{1-\frac{ϕ(m)}{m}}\sim κ_m$ where $κ_m$ is a complicated (unbounded) function of $m$. We show that there exists a positive constant $C(m)$ such that, for all $n \geqslant m$, \[C(m) \left(\frac{n}{m}\right)^{\frac{ϕ(m)}{m}-1} \leqslant ρ(n,m) \leqslant \left(\frac{n}{m}\right)^{\frac{ϕ(m)}{m}-1}\] where $ϕ$ is Euler's totient function.

math.CO↗
arXiv2019

Cyclotomic ordering conjecture

This note describes a conjecture involving cyclotomic polynomials and some initial thoughts towards a solution. Given positive integers $m,n$, the conjecture is that either $Φ_m(q)\leqslantΦ_n(q)$ or $Φ_m(q)\geqslantΦ_n(q)$ holds for all integers $q\geqslant 2$. Pomerance and Rubinstein-Salzedo proved the conjecture in a preprint called `Cyclotomic coincidences' [arXiv:1903.01962].

math.NT↗
arXiv2018

`Norman involutions' and tensor products of unipotent Jordan blocks

A good knowledge of the Jordan canonical form (JCF) for a tensor product of `Jordan blocks' is key to understanding the actions of $p$-groups of matrices in characteristic $p$. The JCF corresponds to a certain partition which depends on the characteristic $p$, and the study of these partitions dates back to Aitken's work in 1934. Equivalently each JCF corresponds to a certain permutation $π$ introduced by Norman in 1995. These permutations $π= π(r,s,p)$ depend on the dimensions $r$, $s$ of the Jordan blocks, and on $p$. We give necessary and sufficient conditions for $π(r,s,p)$ to be trivial, building on work of M.J. Barry. We show that when $π(r,s,p)$ is nontrivial, it is an involution involving reversals. Finally, we prove that the group $G(r,p)$ generated by $π(r,s,p)$ for all $s$, `factors' as a wreath product corresponding to the factorisation $r=ab$ as a product of its $p'$-part $a$ and $p$-part $b$: precisely $G(r, p)={\sf S}_a\wr {\sf D}_b$ where ${\sf S}_a$ is a symmetric group of degree $a$, and ${\sf D}_b$ is a dihedral group of degree $b$.

math.RT↗
arXiv2018

Duality between $p$-groups with three characteristic subgroups and semisimple anti-commutative algebras

Let $p$ be an odd prime and let $G$ be a non-abelian finite $p$-group of exponent $p^2$ with three distinct characteristic subgroups, namely $1$, $G^p$, and $G$. The quotient group $G/G^p$ gives rise to an anti-commutative ${\mathbb F}_p$-algebra $L$ such that the action of ${\rm Aut}(L)$ is irreducible on $L$; we call such an algebra IAC. This paper establishes a duality $G\leftrightarrow L$ between such groups and such IAC algebras. We prove that IAC algebras are semisimple and we classify the simple IAC algebras of dimension at most 4 over certain fields. We also give other examples of simple IAC algebras, including a family related to the $m$-th symmetric power of the natural module of ${\rm SL}(2,{\mathbb F})$.

math.GR↗
arXiv2018

Bounding the composition length of primitive permutation groups and completely reducible linear groups

We obtain upper bounds on the composition length of a finite permutation group in terms of the degree and the number of orbits, and analogous bounds for primitive, quasiprimitive and semiprimitive groups. Similarly, we obtain upper bounds on the composition length of a finite completely reducible linear group in terms of some of its parameters. In almost all cases we show that the bounds are sharp, and describe the extremal examples.

math.GR↗
arXiv2018

On the parameters of intertwining codes

Let $F$ be a field and let $F^{r\times s}$ denote the space of $r\times s$ matrices over $F$. Given equinumerous subsets $\mathcal{A}=\{A_i\mid i \in I\}\subseteq F^{r\times r}$ and $\mathcal{B}=\{B_i\mid i\in I\}\subseteq F^{s\times s}$ we call the subspace $C(\mathcal{A},\mathcal{B}):=\{X\in F^{r\times s}\mid A_iX=XB_i\ {\rm for }\ i\in I\}$ an \emph{intertwining code}. We show that if $C(\mathcal{A},\mathcal{B})\ne\{0\}$, then for each $i\in I$, the characteristic polynomials of $A_i$ and $B_i$ and share a nontrivial factor. We give an exact formula for $k=\dim(C(\mathcal{A},\mathcal{B}))$ and give upper and lower bounds. This generalizes previous work in this area. Finally we construct intertwining codes with large minimum distance when the field is not `too small'. We give examples of codes where $d=rs/k=1/R$ is large where the minimum distance, dimension, and rate of the linear code $C(\mathcal{A},\mathcal{B})$ by $d$, $k$, and $R=k/rs$, respectively.

math.CO↗
arXiv2018

The shape of solvable groups with odd order

We prove that the minimal composition length, $c$, of a solvable group with solvable length $d$ satisfies $9^{(d-3)/9}< c< 9^{(d+1)/5}$, and the minimal composition length, $c^o$, of a group with odd order and solvable length $d$ satisfies $7^{(d-2)/5}< c^o< 2^d$.

math.GR↗
arXiv2017

On the second-largest Sylow subgroup of a finite simple group of Lie type

Let $T$ be a finite simple group of Lie type in characteristic $p$, and let $S$ be a Sylow subgroup of $T$ with maximal order. It is well known that $S$ is a Sylow $p$-subgroup except in an explicit list of exceptions, and that $S$ is always `large' in the sense that $|T|^{1/3} < |S| \leqslant |T|^{1/2}$. One might anticipate that, moreover, the Sylow $r$-subgroups of $T$ with $r \neq p$ are usually significantly smaller than $S$. We verify this hypothesis by proving that for every $T$ and every prime divisor $r$ of $|T|$ with $r \neq p$, the order of the Sylow $r$-subgroup of $T$ at most $|T|^{2\lfloor\log_r(4(\ell+1) r)\rfloor/\ell}=|T|^{{\rm O}(\log_r(\ell)/\ell)}$, where $\ell$ is the Lie rank of $T$.

math.GR↗
arXiv2017

Maximal linear groups induced on the Frattini quotient of a $p$-group

Let $p>3$ be a prime. For each maximal subgroup $H\leqslant\mathrm{GL}(d,p)$ with $|H| \geqslant p^{3d+1}$, we construct a $d$-generator finite $p$-group $G$ with the property that $\mathrm{Aut}(G)$ induces $H$ on the Frattini quotient $G/Φ(G)$ and $|G| \leqslant p^{\frac{d^4}{2}}$. A significant feature of this construction is that $|G|$ is very small compared to $|H|$, shedding new light upon a celebrated result of Bryant and Kovács. The groups $G$ that we exhibit have exponent $p$, and of all such groups $G$ with the desired action of $H$ on $G/Φ(G)$, the construction yields groups with smallest nilpotency class, and in most cases, the smallest order.

math.GR↗
arXiv2017

The number of composition factors of order $p$ in completely reducible groups of characteristic $p$

Let $q$ be a power of a prime $p$ and let $G$ be a completely reducible subgroup of $\mathrm{GL}(d,q)$. We prove that the number of composition factors of $G$ that have prime order $p$ is at most $(\varepsilon_q d-1)/(p-1)$, where $\varepsilon_q$ is a function of $q$ satisfying $1\leqslant\varepsilon_q\leqslant 3/2$. For every $q$, we give examples showing this bound is sharp infinitely often.

math.GR↗
arXiv2017

On the dimension of twisted centralizer codes

Given a field $F$, a scalar $λ\in F$ and a matrix $A\in F^{n\times n}$, the twisted centralizer code $C_F(A,λ):=\{B\in F^{n\times n}\mid AB-λBA=0\}$ is a linear code of length $n^2$. When $A$ is cyclic and $λ\ne0$ we prove that $\dim C_F(A,λ)=\mathrm{deg}(\gcd(c_A(t),λ^n c_A(λ^{-1}t)))$ where $c_A(t)$ denotes the characteristic polynomial of $A$. We also show how $C_F(A,λ)$ decomposes, and we estimate the probability that $C_F(A,λ)$ is nonzero when $|F|$ is finite. Finally, we prove $\dim C_F(A,λ)\leqslant n^2/2$ for $λ\not\in\{0,1\}$ and `almost all' matrices $A$.

math.CO↗
arXiv2017

Twisted Centralizer Codes

Given an $n\times n$ matrix $A$ over a field $F$ and a scalar $a\in F$, we consider the linear codes $C(A,a):=\{B\in F^{n\times n}\mid \,AB=aBA\}$ of length $n^2$. We call $C(A,a)$ a twisted centralizer code. We investigate properties of these codes including their dimensions, minimum distances, parity-check matrices, syndromes, and automorphism groups. The minimal distance of a centralizer code (when $a=1$) is at most $n$, however for $a\ne 0,1$ the minimal distance can be much larger, as large as $n^2$.

math.CO↗
arXiv2016

Arc-transitive digraphs with quasiprimitive local actions

Let $Γ$ be a finite $G$-vertex-transitive digraph. The in-local action of $(Γ,G)$ is the permutation group $L_-$ induced by the vertex-stabiliser on the set of in-neighbours of $v$. The out-local action $L_+$ is defined analogously. Note that $L_-$ and $L_+$ may not be isomorphic. We thus consider the problem of determining which pairs $(L_-,L_+)$ are possible. We prove some general results, but pay special attention to the case when $L_-$ and $L_+$ are both quasiprimitive. (Recall that a permutation group is quasiprimitive if each of its nontrivial normal subgroups is transitive.) Along the way, we prove a structural result about pairs of finite quasiprimitive groups of the same degree, one being (abstractly) isomorphic to a proper quotient of the other.

math.GR↗
arXiv2016

Primitive prime divisors and the $n$-th cyclotomic polynomial

Primitive prime divisors play an important role in group theory and number theory. We study a certain number theoretic quantity, called $Φ^*_n(q)$, which is closely related to the cyclotomic polynomial $Φ_n(x)$ and to primitive prime divisors of $q^n-1$. Our definition of $Φ^*_n(q)$ is novel, and we prove it is equivalent to the definition given by Hering. Given positive constants $c$ and $k$, we give an algorithm for determining all pairs $(n,q)$ with $Φ^*_n(q)\le cn^k$. This algorithm is used to extend (and correct) a result of Hering which is useful for classifying certain families of subgroups of finite linear groups.

math.NT↗