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John R. Britnell

Publications and source records attributed to John R. Britnell.

At least 19 recordsLinked to original sources
arXiv2026

Stallings foldings for rational subsets of automatic groups

Let $G$ be an automatic group with associated regular language $L$. We describe a procedure for constructing an automaton which recognises elements of a given submonoid or rational subset $K$ of $G$. This builds on work of Kharlampovich, Miasnikov and Weil, on the case where $K$ is a subgroup of $G$. Our construction succeeds, after sufficiently many iterations, whenever $K$ satisfies a certain convexity property, which we call $L$-proximity. We show how to test whether the construction is complete in the case that $K$ is a submonoid; we have no such test for the general case of a rational subset $K$. We focus particularly on the case of a surface group $G$ of genus $g>1$, where $L$ is the language of geodesic words in the standard generators. We use small cancellation theory to obtain a method for constructing $L$-recognisable submonoids of $G$.

math.GR↗
arXiv2024

A necessary and sufficient condition for double coset lumping of Markov chains on groups with an application to the random to top shuffle

Let $Q$ be a probability measure on a finite group $G$, and let $H$ be a subgroup of $G$. We show that a necessary and sufficient condition for the random walk driven by $Q$ on $G$ to induce a Markov chain on the double coset space $H\backslash G/H$, is that $Q(gH)$ is constant as $g$ ranges over any double coset of $H$ in $G$. We obtain this result as a corollary of a more general theorem on the double cosets $H \backslash G / K$ for $K$ an arbitrary subgroup of $G$. As an application we study a variation on the $r$-top to random shuffle which we show induces an irreducible, recurrent, reversible and ergodic Markov chain on the relevant double cosets. The transition matrix of the induced walk has remarkable spectral properties: we find its invariant distribution and its eigenvalues and hence determine its rate of convergence.

math.GR↗
arXiv2021

Involutive random walks on total orders and the anti-diagonal eigenvalue property

This paper studies a family of random walks defined on the finite ordinals using their order reversing involutions. Starting at $x \in \{0,1,\ldots,n-1\}$, an element $y \le x$ is chosen according to a prescribed probability distribution, and the walk then steps to $n-1-y$. We show that under very mild assumptions these walks are irreducible, recurrent and ergodic. We then find the invariant distributions, eigenvalues and eigenvectors of a distinguished subfamily of walks whose transition matrices have the global anti-diagonal eigenvalue property studied in earlier work by Ochiai, Sasada, Shirai and Tsuboi. We prove that this subfamily of walks is characterised by their reversibility. As a corollary, we obtain the invariant distributions and rate of convergence of the random walk on the set of subsets of $\{1,\ldots, m\}$ in which steps are taken alternately to subsets and supersets, each chosen equiprobably. We then consider analogously defined random walks on the real interval $[0,1]$ and use techniques from the theory of self adjoint compact operators on Hilbert spaces to prove analogues of the main results in the discrete case.

math.PR↗
arXiv2015

Computing derangement probabilities of the symmetric group acting on k-sets

Let $i(\infty,k)$ be the limiting proportion, as $n \rightarrow \infty$, of permutations in the symmetric group of degree $n$ that fix a $k$-set. We give an algorithm for computing $i(\infty,k)$ and state the values of $i(\infty,k)$ for $k \le 30$. These values are consistent with a conjecture of Peter Cameron that $i(\infty,k)$ is a decreasing function of $k$.

math.CO↗
arXiv2015

Perfect commuting graphs

We classify the finite quasisimple groups whose commuting graph is perfect and we give a general structure theorem for finite groups whose commuting graph is perfect.

math.GR↗
arXiv2015

Bell numbers, partition moves and the eigenvalues of the random-to-top shuffle in Dynkin Types A, B and D

Let $B_t(n)$ be the number of set partitions of a set of size~$t$ into at most $n$ parts and let $B'_t(n)$ be the number of set partitions of $\{1,\ldots, t\}$ into at most $n$ parts such that no part contains both $1$ and~$t$ or both $i$ and $i+1$ for any $i \in \{1,\ldots,t-1\}$. We give two new combinatorial interpretations of the numbers $B_t(n)$ and $B'_t(n)$ using sequences of random-to-top shuffles, %that leave a deck of cards invariant, and sequences of box moves on the Young diagrams of partitions. Using these ideas we obtain a very short proof of a generalization of a result of Phatarfod on the eigenvalues of the random-to-top shuffle. We also prove analogous results for random-to-top shuffles that may flip certain cards. The proofs use the Solomon descent algebras of Types A, B and~D. We give generating functions and asymptotic results for all the combinatorial quantities studied in this paper.

math.CO↗
arXiv2014

Nilpotent covers and non-nilpotent subsets of finite groups of Lie type

Let $G$ be a finite group, and $c$ an element of $\mathbb{Z}\cup \{\infty\}$. A subgroup $H$ of $G$ is said to be {\it $c$-nilpotent} if it is nilpotent, and has nilpotency class at most $c$. A subset $X$ of $G$ is said to be {\it non-$c$-nilpotent} if it contains no two elements $x$ and $y$ such that the subgroup $< x,y>$ is $c$-nilpotent. In this paper we study the quantity $ω_c(G)$, defined to be the size of the largest non-$c$-nilpotent subset of $L$. In the case that $L$ is a finite group of Lie type, we identify covers of $L$ by $c$-nilpotent subgroups, and we use these covers to construct large non-$c$-nilpotent sets in $L$. We prove that for groups $L$ of fixed rank $r$, there exist constants $D_r$ and $E_r$ such that $D_r N \leq ω_\infty(L) \leq E_r N$, where $N$ is the number of maximal tori in $L$. In the case of groups $L$ with twisted rank 1, we provide exact formulae for $ω_c(L)$ for all $c\in\mathbb{Z}\cup \{\infty\}$. If we write $q$ for the level of the Frobenius endomorphism associated with $L$ and assume that $q>5$, then $ω_\infty(G)$ may be expressed as a polynomial in $q$ with coefficients in $\{0,1\}$.

math.GR↗
arXiv2014

On exceptional groups of order p^5

A finite group G is exceptional if it has a quotient Q whose minimal faithful permutation degree is greater than that of G. We say that Q is a distinguished quotient. The smallest examples of exceptional p-groups have order p^5. For an odd prime p, we classify all pairs (G,Q) where G has order p^5 and Q is a distinguished quotient. (The case p=2 has already been treated by Easdown and Praeger.) We establish the striking asymptotic result that as p increases, the proportion of groups of order p^5 with at least one exceptional quotient tends to 1/2.

math.GR↗
arXiv2014

The majority game with an arbitrary majority

The $k$-majority game is played with $n$ numbered balls, each coloured with one of two colours. It is given that there are at least $k$ balls of the majority colour, where $k$ is a fixed integer greater than $n/2$. On each turn the player selects two balls to compare, and it is revealed whether they are of the same colour; the player's aim is to determine a ball of the majority colour. It has been correctly stated by Aigner that the minimum number of comparisons necessary to guarantee success is $2(n-k) - B(n-k)$, where $B(m)$ is the weight of the binary expansion of $m$. However his proof contains an error. We give an alternative proof of this result, which generalizes an argument of Saks and Werman.

math.CO↗
arXiv2013

On types of matrices and centralizers of matrices and permutations

It is known that that the centralizer of a matrix over a finite field depends, up to conjugacy, only on the type of the matrix, in the sense defined by J. A. Green. In this paper an analogue of the type invariant is defined that in general captures more information; using this invariant the result on centralizers is extended to arbitrary fields. The converse is also proved: thus two matrices have conjugate centralizers if and only if they have the same generalized type. The paper ends with the analogous results for symmetric and alternating groups.

math.GR↗
arXiv2012

Normal coverings of linear groups

For a non-cyclic finite group $G$, let $γ(G)$ denote the smallest number of conjugacy classes of proper subgroups of $G$ needed to cover $G$. Bubboloni, Praeger and Spiga, motivated by questions in number theory, have recently established that $γ(S_n)$ and $γ(A_{n})$ are bounded above and below by linear functions of $n$. In this paper we show that if $G$ is in the range $\SL_{n}(q)\le G\le \GL_{n}(q)$ for $n>2$, then $n/π^2 < γ(G) \le (n+1)/2$. We give various alternative bounds, and derive explicit formulas for $γ(G)$ in some cases.

math.GR↗
arXiv2012

Orbit coherence in permutation groups

This paper introduces the notion of orbit coherence in a permutation group. Let $G$ be a group of permutations of a set $Ω$. Let $π(G)$ be the set of partitions of $Ω$ which arise as the orbit partition of an element of $G$. The set of partitions of $Ω$ is naturally ordered by refinement, and admits join and meet operations. We say that $G$ is join-coherent if $π(G)$ is join-closed, and meet-coherent if $π(G)$ is meet-closed. Our central theorem states that the centralizer in $\Sym(Ω)$ of any permutation $g$ is meet-coherent, and subject to a certain finiteness condition on the orbits of $g$, also join-coherent. In particular, if $Ω$ is a finite set then the orbit partitions of elements of the centralizer in $\Sym(Ω)$ of $g$ form a lattice. A related result states that the intransitive direct product and the imprimitive wreath product of two finite permutation groups are join-coherent if and only if each of the groups is join-coherent. We also classify the groups $G$ such that $π(G)$ is a chain and prove two further theorems classifying the primitive join-coherent groups of finite degree, and the join-coherent groups of degree $n$ normalizing a subgroup generated by an $n$-cycle.

math.GR↗
arXiv2012

Finding a princess in a palace: A pursuit-evasion problem

This paper solves a pursuit-evasion problem in which a prince must find a princess who is constrained to move on each day from one vertex of a finite graph to another. Unlike the related and much studied `Cops and Robbers Game', the prince has no knowledge of the position of the princess; he may, however, visit any single room he wishes on each day. We characterize the graphs for which the prince has a winning strategy, and determine, for each such graph, the minimum number of days the prince requires to guarantee to find the princess.

math.CO↗
arXiv2012

The probability that a pair of elements of a finite group are conjugate

Let $G$ be a finite group, and let $κ(G)$ be the probability that elements $g$, $h\in G$ are conjugate, when $g$ and $h$ are chosen independently and uniformly at random. The paper classifies those groups $G$ such that $κ(G) \geq 1/4$, and shows that $G$ is abelian whenever $κ(G)|G| < 7/4$. It is also shown that $κ(G)|G|$ depends only on the isoclinism class of $G$. Specialising to the symmetric group $S_n$, the paper shows that $κ(S_n) \leq C/n^2$ for an explicitly determined constant $C$. This bound leads to an elementary proof of a result of Flajolet \emph{et al}, that $κ(S_n) \sim A/n^2$ as $n\rightarrow \infty$ for some constant $A$. The same techniques provide analogous results for $ρ(S_n)$, the probability that two elements of the symmetric group have conjugates that commute.

math.GR↗
arXiv2010

On types and classes of commuting matrices over finite fields

This paper addresses various questions about pairs of similarity classes of matrices which contain commuting elements. In the case of matrices over finite fields, we show that the problem of determining such pairs reduces to a question about nilpotent classes; this reduction makes use of class types in the sense of Steinberg and Green. We investigate the set of scalars that arise as determinants of elements of the centralizer algebra of a matrix, providing a complete description of this set in terms of the class type of the matrix. Several results are established concerning the commuting of nilpotent classes. Classes which are represented in the centralizer of every nilpotent matrix are classified--this result holds over any field. Nilpotent classes are parametrized by partitions; we find pairs of partitions whose corresponding nilpotent classes commute over some finite fields, but not over others. We conclude by classifying all pairs of classes, parametrized by two-part partitions, that commute. Our results on nilpotent classes complement work of Košir and Oblak.

math.GR↗
arXiv2008

Commuting elements in conjugacy classes: An application of Hall's Marriage Theorem

Let G be a finite group. Define a relation ~ on the conjugacy classes of G by setting C ~ D if there are representatives c \in C and d \in D such that cd = dc. In the case where G has a normal subgroup H such that G/H is cyclic, two theorems are proved concerning the distribution, between cosets of H, of pairs of conjugacy classes of G related by ~. One of the proofs involves an interesting application of the famous Marriage Theorem of Philip Hall. The paper concludes by discussing some aspects of these theorems and of the relation ~ in the particular cases of symmetric and general linear groups, and by mentioning an open question related to Frobenius groups.

math.GR↗
arXiv2007

On the distribution of conjugacy classes between the cosets of a finite group in a cyclic extension

Let G be a finite group and H a normal subgroup such that G/H is cyclic. Given a conjugacy class g^G of G we define its centralizing subgroup to be HC_G(g). Let K be such that H\le K\le G. We show that the G-conjugacy classes contained in K whose centralizing subgroup is K, are equally distributed between the cosets of H in K. The proof of this result is entirely elementary. As an application we find expressions for the number of conjugacy classes of K under its own action, in terms of quantities relating only to the action of G.

math.GR↗