arXiv subjects
Publications and source records attributed to Nick Gill.
We prove Cherlin's conjecture, concerning binary primitive permutation groups, for those groups with socle isomorphic to $\mathrm{PSL}_2(q)$, ${^2\mathrm{B}_2}(q)$, ${^2\mathrm{G}_2}(q)$ or $\mathrm{PSU}_3(q)$. Our method uses the notion of a "strongly non-binary action".
We introduce a new approach to the study of finite binary permutation groups and, as an application of our method, we prove Cherlin's binary groups conjecture for groups with socle a finite alternating group, and for the $\mathcal{C}_1$-primitive actions of the finite classical groups. Our new approach involves the notion, defined with respect to a group action, of a `\emph{beautiful subset}'. We demonstrate how the presence of such subsets can be used to show that a given action is not binary. In particular, the study of such sets will lead to a resolution of many of the remaining open cases of Cherlin's binary groups conjecture.
In 1997, John Conway constructed a $6$-fold transitive subset $M_{13}$ of permutations on a set of size $13$ for which the subset fixing any given point was isomorphic to the Mathieu group $M_{12}$. The construction was via a "moving-counter puzzle" on the projective plane ${\rm PG}(2,3)$. We discuss consequences and generalisations of Conway's construction. In particular we explore how various designs and hypergraphs can be used instead of ${\rm PG}(2,3)$ to obtain interesting analogues of $M_{13}$; we refer to these analogues as Conway groupoids. A number of open questions are presented.
To a set $\mathcal{B}$ of 4-subsets of a set $Ω$ of size $n$ we introduce an invariant called the `hole stabilizer' which generalises a construction of Conway, Elkies and Martin of the Mathieu group $M_{12}$ based on Loyd's `15-puzzle'. It is shown that hole stabilizers may be regarded as objects inside an objective partial group (in the sense of Chermak). We classify pairs $(Ω,\mathcal{B})$ with a trivial hole stabilizer, and determine all hole stabilizers associated to $2$-$(n,4,λ)$ designs with $λ\leq 2$.
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.
A $2-(n,4,λ)$ design $(Ω, \mathcal{B})$ is said to be supersimple if distinct lines intersect in at most two points. From such a design, one can construct a certain subset of Sym$(Ω)$ called a "Conway groupoid". The construction generalizes Conway's construction of the groupoid $M_{13}$. It turns out that several infinite families of groupoids arise in this way, some associated with 3-transposition groups, which have two additional properties. Firstly the set of collinear point-triples forms a regular two-graph, and secondly the symmetric difference of two intersecting lines is again a line. In this paper, we show each of these properties corresponds to a group-theoretic property on the groupoid and we classify the Conway groupoids and the supersimple designs for which both of these two additional properties hold.
In this note we use a result of Kantor to prove a conjecture of Degos. Specifically we prove the following: let $\mathbb{F}$ be a finite field of order $q$ and let $f, g\in\mathbb{F}[X]$ be distinct polynomials of degree $n$ such that $f$ is primitive, and the constant term of $g$ is non-zero. Then $ =\mathrm{GL}_n(q)$.
Let $G=A_n$, a finite alternating group. We study the commuting graph of $G$ and establish, for all possible values of $n$ barring $13, 14, 17$ and $19$, whether or not the independence number is equal to the clique-covering number.
To each supersimple $2-(n,4,λ)$ design $\mathcal{D}$ one associates a `Conway groupoid,' which may be thought of as a natural generalisation of Conway's Mathieu groupoid associated to $M_{13}$ which is constructed from $\mathbb{P}_3$. We show that $\operatorname{Sp}_{2m}(2)$ and $2^{2m}.\operatorname{Sp}_{2m}(2)$ naturally occur as Conway groupoids associated to certain designs. It is shown that the incidence matrix associated to one of these designs generates a new family of completely transitive $\mathbb{F}_2$-linear codes with minimum distance 4 and covering radius 3, whereas the incidence matrix of the other design gives an alternative construction to a previously known family of completely transitive codes. We also give a new characterization of $M_{13}$ and prove that, for a fixed $λ> 0,$ there are finitely many Conway groupoids for which the set of morphisms does not contain all elements of the full alternating or symmetric group.
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 $\{0,1\}$.
Suppose that a group $G$ acts transitively on the points of $\mathcal{P}$, a finite non-Desarguesian projective plane. We prove that if $G$ is insoluble then $G/O(G)$ is isomorphic to $SL_2(5)$ or $SL_2(5).2$.
Let $K=Z/pZ$ and let $A$ be a subset of $\GL_r(K)$ such that $ $ is solvable. We reduce the study of the growth of $A$ under the group operation to the nilpotent setting. Specifically we prove that either $A$ grows rapidly (meaning $|A\cdot A\cdot A|\gg |A|^{1+δ}$), or else there are groups $U_R$ and $S$, with $S/U_R$ nilpotent such that $A_k\cap S$ is large and $U_R\subseteq A_k$, where $k$ is a bounded integer and $A_k = \{x_1 x_2...b x_k : x_i \in A \cup A^{-1} \cup {1}}$. The implied constants depend only on the rank $r$ of $\GL_r(K)$. When combined with recent work by Pyber and Szabó, the main result of this paper implies that it is possible to draw the same conclusions without supposing that $ $ is solvable.
Let $G$ be a finite group acting vertex-transitively on a graph. We show that bounding the order of a vertex stabilizer is equivalent to bounding the second singular value of a particular bipartite graph. This yields an alternative formulation of the Weiss Conjecture.
Let $G$ be a finite group acting transitively on a set $Ω$. We study what it means for this action to be {\it quasirandom}, thereby generalizing Gowers' study of quasirandomness in groups. We connect this notion of quasirandomness to an upper bound for the convolution of functions associated with the action of $G$ on $Ω$. This convolution bound allows us to give sufficient conditions such that sets $S,T\subset G$ and $Γ\subseteq Ω$ contain elements $s\in S, t\in T, γ\inΓ$ such that $s(γ)=t$. Other consequences include an analogue of `the Gowers trick' of Nikolov and Pyber for general group actions, a sum-product type theorem for large subsets of a finite field, as well as applications to expanders and to the study of the diameter and width of a finite simple group.
There is a parallelism between growth in arithmetic combinatorics and growth in a geometric context. While, over $\mathbb{R}$ or $\mathbb{C}$, geometric statements on growth often have geometric proofs, what little is known over finite fields rests on arithmetic proofs. We discuss strategies for geometric proofs of growth over finite fields, and show that growth can be defined and proven in an abstract projective plane -- even one with weak axioms.
We prove that if $G$ is a finite simple group of Lie type and $S$ a subset of $G$ of size at least two then $G$ is a product of at most $c\log|G|/\log|S|$ conjugates of $S$, where $c$ depends only on the Lie rank of $G$. This confirms a conjecture of Liebeck, Nikolov and Shalev in the case of families of simple groups of Lie type of bounded rank.
We study those $(2,m,n)$-groups which are almost simple and for which the absolute value of the Euler characteristic is a product of two prime powers. All such groups which are not isomorphic to $PSL_2(q)$ or $PGL_2(q)$ are completely classified.
Let $G$ be a $(2,m,n)$-group and let $x$ be the number of distinct primes dividing $χ$, the Euler characteristic of $G$. We prove, first, that, apart from a finite number of known exceptions, a non-abelian simple composition factor $T$ of $G$ is a finite group of Lie type with rank $n\leq x$. This result is proved using new results connecting the prime graph of $T$ to the integer $x$. We then study the particular cases $x=1$ and $x=2$. We give a general structure statement for $(2,m,n)$-groups which have Euler characteristic a prime power, and we construct an infinite family of these objects. We also give a complete classification of those $(2,m,n)$-groups which are almost simple and for which the Euler characteristic is a prime power (there are four such). Finally we study those $(2,m,n)$-groups which are almost simple and for which the Euler characteristic is a product of two prime powers. All such groups which are not isomorphic to $PSL_2(q)$ or $PGL_2(q)$ are completely classified.