The mathematics of shuffling a deck of $2n$ cards with two "perfect shuffles" was brought into clarity by Diaconis, Graham and Kantor. Here we consider a generalisation of this problem, with a so-called "many handed dealer" shuffling $kn$ cards by cutting into $k$ piles with $n$ cards in each pile and using $k!$ shuffles. A conjecture of Medvedoff and Morrison suggests that all possible permutations of the deck of cards are achieved, so long as $k\neq 4$ and $n$ is not a power of $k$. We confirm this conjecture for three doubly infinite families of integers: all $(k,n)$ with $k>n$; all $(k, n)\in \{ (\ell^e, \ell^f )\mid \ell \geqslant 2, \ell^e>4, f \ \mbox{not a multiple of}\ e\}$; and all $(k,n)$ with $k=2^e\geqslant 4$ and $n$ not a power of $2$. We open up a more general study of shuffle groups, which admit an arbitrary subgroup of shuffles.
For a prime number $p$ and a free profinite group $S$ on the basis $X$, let $S_{(n,p)}$, $n=1,2,\ldots,$ be the $p$-Zassenhaus filtration of $S$. For $p>n$, we give a word-combinatorial description of the cohomology group $H^2(S/S_{(n,p)},\mathbb{Z}/p)$ in terms of the shuffle algebra on $X$. We give a natural linear basis for this cohomology group, which is constructed by means of unitriangular representations arising from Lyndon words.
Standard perfect shuffles involve splitting a deck of $2n$ cards into two stacks and interlacing the cards from the stacks. There are two ways that this interlacing can be done, commonly referred to as an in shuffle and an out shuffle, respectively. In 1983, Diaconis, Graham, and Kantor determined the permutation group generated by in and out shuffles on a deck of $2n$ cards for all $n$. Diaconis et al. concluded their work by asking whether similar results can be found for so-called generalized perfect shuffles. For these new shuffles, we split a deck of $mn$ cards into $m$ stacks and similar…
The generalised Fitting subgroup of a finite group is the group generated by all subnormal subgroups that are either nilpotent or quasisimple. The importance of this subgroup in finite group theory stems from the fact that it always contains its own centraliser, so that any finite group is an abelian extension of a group of automorphisms of its generalised Fitting subgroup. We define a class of profinite groups which generalises this phenomenon, and explore some consequences for the structure of profinite groups.
We study centralisers of finite order automorphisms of generalisations of Thompson's group F and conjugacy classes of finite subgroups in finite extensions of these groups. In particular, we show that centralisers of finite automorphisms in these generalised Thompson groups either admit a finite type classifying space or are not finitely generated. As an application we deduce results about the Bredon type of such finite extensions.
We prove that the generalised word problem of a finitely generated subgroup of a finitely generated virtually free group is context-free, that a hyperbolic group must be virtually free if it has a torsion-free quasiconvex subgroup of infinite index with context-free generalised word problem, and that, for any hyperbolic group, the generalised word problem of a torsion-free quasiconvex subgroup is recognised by a real-time Turing machine.
In this article, we completely characterize the asymptotic behavior of conjugacy separability for the lamplighter groups. More generally, we give exponential upper and lower bounds for all wreath products of finitely generated abelian groups where the acting group is infinite and base group is finite.
Let $p\ge 3$ be a prime. A generalised multi-edge spinal group is a subgroup of the automorphism group of a regular $p$-adic rooted tree T that is generated by one rooted automorphism and $p$ families of directed automorphisms, each family sharing a common directed path disjoint from the paths of the other families. This notion generalises the concepts of multi-edge spinal groups, including the widely studied GGS-groups, and the extended Gupta-Sidki groups that were introduced by Pervova. Extending techniques that were developed in these more special cases, we prove: generalised multi-edge spi…
If G is a group with a presentation of the form < x,y|x^3=y^5=W(x,y)^2=1 >, then either G is virtually soluble or G contains a free subgroup of rank 2. This provides additional evidence in favour of a conjecture of Rosenberger.
We prove that, for a finitely generated group hyperbolic relative to virtually abelian subgroups, the generalised word problem for a parabolic subgroup is the language of a real-time Turing machine. Then, for a hyperbolic group, we show that the generalised word problem for a quasiconvex subgroup is a real-time language under either of two additional hypotheses on the subgroup. By extending the Muller-Schupp theorem we show that the generalised word problem for a finitely generated subgroup of a finitely generated virtually free group is context-free. Conversely, we prove that a hyperbolic gro…
If G is a group with a presentation of the form < x,y|x^3=y^3=W(x,y)^2=1 >, then either G is virtually soluble or G contains a free subgroup of rank 2. This provides additional evidence in favour of a conjecture of Rosenberger.
For an arbitrary sequence $(G_s)$ of subgroups of finite index in the generalised Thompson group $$F_{n, \infty} = \langle x_0, x_1, \ldots, x_m, \ldots \mid x_i^{x_j} = x_{i+ n-1} \hbox{ for } i > j \geq 0 \rangle$$ it is shown that $\sup_{s \geq 1} d(G_s) < \infty$ and that the deficiency gradient of $F_{n, \infty}$ with respect to $(G_s)$ is 0 provided $[G : G_s]$ tends to infinity. A higher dimensional analogue is considered for $n = 2$.
Brown has defined the generalised Thompson's group $F_n$, $T_n$, where $n$ is an integer at least $2$ and Thompson's groups $F= F_2$ and $T =T_2$ in the 80's. Burillo, Cleary and Stein have found that there is a quasi-isometric embedding from $F_n$ to $F_m$ where $n$ and $m$ are positive integers at least 2. We show that there is a quasi-isometric embedding from $T_n$ to $T_2$ for any $n \geq 2$ and no embeddings from $T_2$ to $T_n$ for $n \geq 3$.
Let $V_r(Σ)$ be the generalised Thompson group defined as the automorphism group of a valid, bounded, and complete Cantor algebra. We show that that for every $n>0$ there is a $k>n,$ such that there exists a $k$-dimensional $n$-connected simplicial complex $K$ such that $V_r(Σ)$ acts on $K$ with finite stabilisers. We also determine the number of conjugacy classes of finite cyclic subgroups of a given order $m$ in Brin-Thompson groups. We apply our computations to the rationalised Farrell-Jones assembly map in algebraic $K$-theory.
We study a family of hereditarily just infinite profinite groups obtained by iterated wreath products introduced by J. Wilson in 2010. We find explicit generators for this family in a number of cases using combinatorial methods. We then discuss determination of the lower rank for this family. Finally we examine whether the groups in this family admit a finite profinite presentation.
Golan and Sapir \cite{MR3978542} proved that the Thompson's groups $F$, $T$ and $V$ have linear divergence. In the current paper, we focus on the divergence properties of several generalisation of the Thompson's groups, we first consider the Brown-Thompson's groups $F_n$, $T_n$ and $V_n$ and found out that these groups also have linear divergence function. We then consider the braided Thompson's groups $BF$ and $\widehat{BF}$ and $\widehat{BV}$ together with the result in \cite{Kodama:2020to} we conclude that theses groups have linear divergence.
Let $G$ be a collineation group of a thick finite generalised hexagon or generalised octagon $Γ$. If $G$ acts primitively on the points of $Γ$, then a recent result of Bamberg et al. shows that $G$ must be an almost simple group of Lie type. We show that, furthermore, the minimal normal subgroup $S$ of $G$ cannot be a Suzuki group or a Ree group of type $^2\text{G}_2$, and that if $S$ is a Ree group of type $^2\text{F}_4$, then $Γ$ is (up to point--line duality) the classical Ree--Tits generalised octagon.
In this paper, we find a strong new restriction on the structure of CI-groups. We show that, if $R$ is a generalised dihedral group and if $R$ is a CI-group, then for every odd prime $p$ the Sylow $p$-subgroup of $R$ has order $p$, or $9$. Consequently, any CI-group with quotient a generalised dihedral group has the same restriction, that for every odd prime $p$ the Sylow $p$-subgroup of the group has order $p$, or $9$. We also give a counter example to the conjecture that every BCI-group is a CI-group.
We prove that splendid Morita equivalences between principal blocks of finite groups with dihedral Sylow $2$-subgroups realised by Scott modules can be lifted to splendid Morita equivalences between principal blocks of finite groups with generalised quaternion Sylow $2$-subgroups realised by Scott modules.
A generalised quadrangle is a point-line incidence geometry G such that: (i) any two points lie on at most one line, and (ii) given a line L and a point p not incident with L, there is a unique point on L collinear with p. They are a specific case of the generalised polygons introduced by Tits, and these structures and their automorphism groups are of some importance in finite geometry. An integral part of understanding the automorphism groups of finite generalised quadrangles is knowing which groups can act primitively on their points, and in particular, which almost simple groups arise as au…
A Beauville group acts freely on the product of two compact Riemann surfaces and faithfully on each one of them. In this paper, we consider higher products and present {\it{generalised Beauville groups}}: for $d \geq 2$, $d$ is the minimal value for which the same action can be defined on the product of $d$ compact Riemann surfaces.
A graph is called a GRR if its automorphism group acts regularly on its vertex-set. Such a graph is necessarily a Cayley graph. Godsil has shown that there are only two infinite families of finite groups that do not admit GRRs : abelian groups and generalised dicyclic groups. Indeed, any Cayley graph on such a group admits specific additional graph automorphisms that depend only on the group. Recently, Dobson and the last two authors showed that almost all Cayley graphs on abelian groups admit no automorphisms other than these obvious necessary ones. In this paper, we prove the analogous resul…
We discuss recent progress on the problem of classifying point-primitive generalised polygons. In the case of generalised hexagons and generalised octagons, this has reduced the problem to primitive actions of almost simple groups of Lie type. To illustrate how the natural geometry of these groups may be used in this study, we show that if $\mathcal{S}$ is a finite thick generalised hexagon or octagon with $G \leqslant{\rm Aut}(\mathcal{S})$ acting point-primitively and the socle of $G$ isomorphic to ${\rm PSL}_n(q)$ where $n \geqslant 2$, then the stabiliser of a point acts irreducibly on the…
In 2008, Schneider and Van Maldeghem proved that if a group acts flag-transitively, point-primitively, and line-primitively on a generalised hexagon or generalised octagon, then it is an almost simple group of Lie type. We show that point-primitivity is sufficient for the same conclusion, regardless of the action on lines or flags. This result narrows the search for generalised hexagons or octagons with point- or line-primitive collineation groups beyond the classical examples, namely the two generalised hexagons and one generalised octagon admitting the Lie type groups $\mathsf{G}_2(q)$, $\,^…
We prove that the generalised non-crossing partitions associated to well-generated complex reflection groups of exceptional type obey two different cyclic sieving phenomena, as conjectured by Armstrong, respectively by Bessis and Reiner. This manuscript accompanies the paper "Cyclic sieving for generalised non-crossing partitions associated to complex reflection groups of exceptional type" [arXiv:1001.0028], for which it provides the computational details.
We establish upper and lower bounds on the dimension of the space spanned by the symmetric powers of the natural character of generalised symmetric groups. We adapt the methods of Savitt and Stanley from their paper `A note on the symmetric powers of the standard representation of $S_n$' (The Electronic Journal of Combinatorics 7 (2000), R6) to obtain bounds both over the complex numbers and in prime characteristic.
We present the proof of the cyclic sieving conjectures for generalised non-crossing partitions associated to well-generated complex reflection groups due to Armstrong, respectively to Bessis and Reiner, for the 26 exceptional well-generated complex reflection groups. The computational details are provided in the manuscript "Cyclic sieving for generalised non-crossing partitions associated to complex reflection groups of exceptional type - the details" [arXiv:1001.0030].
The classification of flag-transitive generalised quadrangles is a long-standing open problem at the interface of finite geometry and permutation group theory. Given that all known flag-transitive generalised quadrangles are also point-primitive (up to point-line duality), it is likewise natural to seek a classification of the point-primitive examples. Working towards this aim, we are led to investigate generalised quadrangles that admit a collineation group $G$ preserving a Cartesian product decomposition of the set of points. It is shown that, under a generic assumption on $G$, the number of…
The generalised Paley graphs are, as their name suggests, a generalisation of the Paley graphs, first defined by Paley in 1933 (see \cite{Paley}). They arise as the relation graphs of symmetric cyclotomic association schemes. However, their automorphism groups may be much larger than the groups of the corresponding schemes. We determine the parameters for which the graphs are connected, or equivalently, the schemes are primitive. Also we prove that generalised Paley graphs are sometimes isomorphic to Hamming graphs and consequently have large automorphism groups, and we determine precisely the…
Külshammer, Olsson, and Robinson developed an l-analogue of modular representation theory of symmetric groups where l is not necessarily a prime. They gave a conjectural combinatorial description for invariant factors of the Cartan matrix in this context. We confirm their conjecture by proving a more precise blockwise conjecture due to Bessenrodt and Hill.
Let G be a connected reductive group and X an equivariant compactifiction of G. In X, we study generalised and opposite generalised Schubert varieties, their intersections called generalised Richardson varieties and projected generalised Richardson varieties. Any complete G-embedding has a canonical Frobenius splitting and we prove that the compatibly split subvarieties are the generalised projected Richardson varieties extending a result of Knutson, Lam and Speyer to the situation.
We have reviewed some results on quantized shuffling, and in particular, the grading and structure of this algebra. In parallel, we have summarized certain details about classical shuffle algebras, including Lyndon words (primes) and the construction of bases of classical shuffle algebras in terms of Lyndon words. We have explained how to adapt this theory to the construction of bases of quantum group algebras in terms of Lyndon words. This method has a limited application to the specific case of the quantum group parameter being a root of unity, with the requirement that specialization to the…
We verify the Generalised Moonshine conjectures for some irrational modular functions for the Monster centralisers related to the Harada-Norton, Held, $M_{12}$ and $L_3(3)$ simple groups based on certain orbifolding constraints. We find explicitly the fixing groups of the hauptmoduls arising in each case.
This note describes an application of the theory of generalised Burnside rings to algebraic representation theory. Tables of marks are given explicitly for the groups $S_4$ and $S_5$ which are of particular interest in the context of reductive algebraic groups. As an application, the base sets for the nilpotent element $F_4 (a_3)$ are computed.
Tent and Ziegler proved that the automorphism group of the Urysohn sphere is simple and that the automorphism group of the Urysohn space is simple modulo bounded automorphisms. A key component of their proof is the definition of a stationary independence relation (SIR). In this paper we prove that the existence of a SIR satisfying some extra axioms is enough to prove simplicity of the automorphism group of a countable structure. The extra axioms are chosen with applications in mind, namely homogeneous structures which admit a "metric-like amalgamation", for example all primitive 3-constrained …
We consider orbifoldings of the Moonshine Module with respect to the abelian group generated by a pair of commuting Monster group elements with one of prime order $p=2,3,5,7$ and the other of order $pk$ for $k=1$ or $k$ prime. We show that constraints arising from meromorphic orbifold conformal field theory allow us to demonstrate that each orbifold partition function with rational coefficients is either constant or is a hauptmodul for an explicitly found modular fixing group of genus zero. We thus confirm in the cases considered the Generalised Moonshine conjectures for all rational modular f…
