Derek F. Holt
The history, definition and principal properties of automatic groups and their generalisations to subgroups and cosets are reviewed briefly, mainly from a computational perspective. A result about the asynchronous automaticity of an HNN extension is then proved and applied to an example that was proposed by Mark Sapir.
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We consider the coset poset associated with the families of proper subgroups, proper subgroups of finite index, and proper normal subgroups of finite index. We investigate under which conditions those coset posets have contractible geometric realizations.
Morse and stable subgroups via the coset intersection complex
本文在陪集交集复形框架下，证明了在群陪集交集复形的特定条件下，无限指标的Morse子群是稳定的，从而建立了Morse子群与稳定子群的等价性。
Double coset problem for parabolic subgroups of braid groups
针对辫群中所有具有连通Coxeter图的抛物子群，首次给出双陪集问题的解，并推广至所有抛物子群。
Let w be a group-word. Suppose that the set of all w-values in a profinite group G is contained in a union of countably many cosets of subgroups. We are concerned with the question to what extent the structure of the verbal subgroup w(G) depends on the properties of the subgroups. We prove the following theorem. Let C be a class of groups closed under taking subgroups, quotients, and such that in any group the product of finitely many normal C-subgroups is again a C-subgroup. If w is a multilinear commutator and G is a profinite group such that the set of all w-values is contained in a union o…
On the number and sizes of double cosets of Sylow subgroups of the symmetric group
研究对称群 Sylow $p$-子群双陪集的数量与大小，证明 $p$ 为奇数且 $n$ 较大时大多数双陪集达最大规模，并发现除少数小例外所有可能大小均出现。
We describe an algorithm that computes the index of a finitely generated subgroup in a finitely $L$-presented group provided that this index is finite. This algorithm shows that the subgroup membership problem for finite index subgroups in a finitely $L$-presented group is decidable. As an application, we consider the low-index subgroups of some self-similar groups including the Grigorchuk group, the twisted twin of the Grigorchuk group, the Grigorchuk super-group and the Hanoi 3-group
Complete algorithms for transversals and double coset representatives of subgroups via subfactors
本文证明了获取子群所有陪集代表系的完备算法，并提出一种通用算法以获取子群所有双重陪集代表系及其数量，同时引入了三个子集的中间直积概念及获取中间因子的方法。
Motivated by questions arising in connection with branched coverings of connected smooth projective curves over finite fields, we study the proportion of fixed point free elements (derangements) in cosets of normal subgroups of primitive permutations groups. Using the Aschbacher-O'Nan-Scott theorem for primitive groups to partition the problem, we provide complete answers for affine groups and groups which contain a regular normal nonabelian subgroup.
Coset complexes of $p$-subgroups in finite groups
本文研究有限群中 p-子群陪集偏序集 C_p(G) 的几何实现，受 Brown 问题启发刻画了 p-闭群的同伦型，并证明其欧拉示性数满足 χ(C_p(G)) ≡ |G|_{p'} (mod p)。
This paper presents a systematic operator theory approach for abstract structure of Banach measure algebras over coset spaces of compact subgroups. Let $H$ be a compact subgroup of a locally compact group $G$ and $G/H$ be the left coset space associated to the subgroup $H$ in $G$. Also, let $M(G/H)$ be the Banach measure space consists of all complex measures over $G/H$. We then introduce an operator theoretic characterization for the abstract notion of involution over the Banach measure space $M(G/H)$.
Cosets of normal subgroups and union of two conjugacy classes
研究有限群正规子群陪集包含于两个共轭类并集的条件，证明此时 $N$ 不必可解但结构受限，其非交换主因子同构于奇特征 Lie 型单群的直积。
We study the connectivity of the coset poset and the subgroup poset of a group, focusing in particular on simple connectivity. The coset poset was recently introduced by K. S. Brown in connection with the probabilistic zeta function of a group. We further Brown's study of the homotopy type of the coset poset, and in particular generalize his results on direct products and classify direct products with simply connected coset posets. The homotopy type of the subgroup poset L(G) has been examined previously by Kratzer, Thevenaz, and Shareshian. We generalize some results of Kratzer and Thevenaz, …
Let $G$ be a classical algebraic group, $X$ a maximal rank reductive subgroup and $P$ a parabolic subgroup. This paper classifies when $X\G/P$ is finite. Finiteness is proven using geometric arguments about the action of $X$ on subspaces of the natural module for $G$. Infiniteness is proven using a dimension criterion which involves root systems.
In a previous article by the author and P. Wesolek, it was shown that a compactly generated locally compact group $G$ admits a finite normal series $(G_i)$ in which the factors are compact, discrete or irreducible in the sense that no closed normal subgroup of $G$ lies properly between $G_{i-1}$ and $G_{i}$. In the present article, we generalize this series to an analogous decomposition of the coset space $G/H$ with respect to closed subgroups, where $G$ is locally compact and $H$ is compactly generated. This time, the irreducible factors are coset spaces $G_{i}/G_{i-1}$ where $G_{i}$ is compa…
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.
We prove that for any prime p there exist infinitely many finite simple groups G with a coset xP of a Sylow p-subgroup P of G such that every element of xP has order divisible by p. John Thompson proved this for p=2 in 1967 answering a question of Lowell Paige. This result is used to answer a question of Richard Taylor on adequate representations.
We study some properties of the coset poset associated with the family of subgroups of class $\leq 2$ of a nilpotent group of class $\leq 3$. We prove that under certain assumptions on the group the coset poset is simply-connected if and only if the group is $2$-Engel, and $2$-connected if and only if the group is nilpotent of class $2$ or less. We determine the homotopy type of the coset poset for the group of $4\times 4$ upper unitriangular matrices over $\mathbb{F}_p$, and for the Burnside groups of exponent $3$.
If a soluble group $G$ contains two finitely generated abelian subgroups $A,B$ such that the number of double cosets $AgB$ is finite, then $G$ is shown to be virtually polycyclic.
Product separability in central extensions
本文研究了中心扩张在子群可分离性和双余集可分离性下的性质，证明了在特定条件下中心扩张的乘积可分离性和双余集可分离性的等价性及稳定性。
Let H, K be subgroups of G. We investigate the intersection properties of left and right cosets of these subgroups.
We characterize which permutational wreath products W^(X)\rtimes G are finitely presented. This occurs if and only if G and W are finitely presented, G acts on X with finitely generated stabilizers, and with finitely many orbits on the cartesian square X^2. On the one hand, this extends a result of G. Baumslag about standard wreath products; on the other hand, this provides nontrivial examples of finitely presented groups. For instance, we obtain two quasi-isometric finitely presented groups, one of which is torsion-free and the other has an infinite torsion subgroup. Motivated by the characte…
We parametrize the space of double cosets of the group $GL(n,\Bbbk)$ with respect to two subgroups $T_-$, $T_+$ of block strictly triangular matrices. In Addendum, we consider the quasi-regular representation of $GL(n,\Bbb{C})$ in $L^2$ on $T_- \ GL(n,\Bbb{C})$, observe that it admits an additional group of symmetries, find the joint spectrum, and observe that it is multiplicity free.
Let $G$ be a totally disconnected, locally compact group and let $H$ be an equicontinuously (for example, compactly) generated group of automorphisms of $G$. We show that every distal action of $H$ on a coset space of $G$ is a SIN action, with the small invariant neighbourhoods arising from open $H$-invariant subgroups. We obtain a number of consequences for the structure of the collection of open subgroups. For example, it follows that for every compactly generated subgroup $K$ of $G$, there is a compactly generated open subgroup $E$ of $G$ such that $K \le E$ and such that every open subgrou…
Following Babai's algorithm for the string isomorphism problem, we determine that it is possible to write expressions of short length describing certain permutation cosets, including all permutation subgroups; this is feasible both in the original version of the algorithm and in its CFSG-free version, partially done by Babai and completed by Pyber. The existence of such descriptions gives a weak form of the Cameron-Maróti classification even without assuming CFSG. We also thoroughly explicate Babai's recursion process (as given in Helfgott) and obtain explicit constants for the runtime of the …
This paper deals with combinatorial aspects of finite covers of groups by cosets or subgroups. Let $a_1G_1,...,a_kG_k$ be left cosets in a group $G$ such that ${a_iG_i}_{i=1}^k$ covers each element of $G$ at least $m$ times but none of its proper subsystems does. We show that if $G$ is cyclic, or $G$ is finite and $G_1,...,G_k$ are normal Hall subgroups of $G$, then $k\geq m+f([G:\bigcap_{i=1}^kG_i])$, where $f(\prod_{t=1}^r p_t^{α_t})=\sum_{t=1}^rα_t(p_t-1)$ if $p_1,...,p_r$ are distinct primes and $α_1,...,α_r$ are nonnegative integers. When all the $a_i$ are the identity element of $G$ and …
This paper studies whether there are distance-transitive graphs arising from the coset actions of $G_2(q)$ on the subfield subgroup $G_2(\sqrt{q})$ or $G_2(q)$ on the Ree subgroup $^2G_2(q)$. It is found that there are no such graphs, even if the groups are extended by outer automorphisms of $G_2(q)$.
Let $H$ be a hyperbolic group, $A$ and $B$ be subgroups of $H$, and $gr(H,A,B)$ be the growth function of the double cosets $AhB, h \in H$. We prove that the behavior of $gr(H,A,B)$ splits into two different cases. If $A$ and $B$ are not quasiconvex, we obtain that every growth function of a finitely presented group can appear as $gr(H,A,B)$. We can even take $A=B$. In contrast, for quasiconvex subgroups A and B of infinite index, $gr(H,A,B)$ is exponential. Moreover, there exists a constant $λ> 0$, such that $gr(H,A,B)(r) >λf_H(r)$ for all big enough $r$, where $f_H(r)$ is the growth function…
In this paper, we show that any subgroup of a semidirect product of Z^n with Z has bounded packing as long as the action of Z on Z^n is by diagonalizable automorphisms all of whose eigenvalues are real. We use this result to show that any subgroup in a polycyclic group of length 3 or less has bounded packing. We also introduce the notion of coset growth and obtain a bound for the coset growth of subgroup H=<t> in the semidirect product of Z^2 with Z.
Large products of double cosets for symmetric subgroups
研究分类满足KxKyK=G的x,y对，给出一般简单紧连通李群G的必要条件，并在G=SU(n)时完全分类，排除AIII类型子群。应用于量子计算门分解。
We prove that the Herzog-Schönheim Conjecture holds for any group $G$ of order smaller than $1440$. In other words we show that in any non-trivial coset partition $\{g_i U_i\}_{i=1}^n $ of $G$ there exist distinct $1 \leq i, j \leq n$ such that $[G:U_i]=[G:U_j]$. We also study interaction between the indices of subgroups having cosets with pairwise trivial intersection and harmonic integers. We prove that if $U_1$,...,$U_n$ are subgroups of $G$ which have pairwise trivially intersecting cosets and $n \leq 4$ then $[G:U_1]$,...,$[G:U_n]$ are harmonic integers.
Let $H$ and $K$ be subgroups of a finite group $G$. Pick $g \in G$ uniformly at random. We study the distribution induced on double cosets. Three examples are treated in detail: 1) $H = K = $ the Borel subgroup in $GL_n(\mathbb{F}_q)$. This leads to new theorems for Mallows measure on permutations and new insights into the LU matrix factorization. 2) The double cosets of the hyperoctahedral group inside $S_{2n}$, which leads to new applications of the Ewens's sampling formula of mathematical genetics. 3) Finally, if $H$ and $K$ are parabolic subgroups of $S_n$, the double cosets are `contingen…
On the complement of a union of cosets
本文证明任意群中 $n$ 个陪集并集的补集可由至多 $2^n$ 个左平移覆盖，有限情形给出下界并解决猜想，同时改进 Neumann 引理。
Let $(W,S)$ be a Coxeter system of finite rank and let $J,K\subset S$. We study the rationality of the Poincaré series of the set of representatives of minimal length of $(W_J,W_K)$-double cosets of $W$: we conclude that it depends mostly on the rationality of the Poincaré series of the normalizers of finite parabolic subgroups of $W$. For affine Weyl groups, we prove that all these series are rational and we give some explicit examples.
It is shown that for any finitely generated subgroups H and K of a free group F, and for any element g in F the double coset HgK is closed in the profinite topology of F.
Subgroup and Coset Intersection in abelian-by-cyclic groups
本文研究无限群中的子群交与陪集交判定问题，通过将二者归约为 Shifted Monomial Membership 问题，证明其在有限生成 abelian-by-cyclic 群中可判定，并讨论推广到任意 metabelian 群的障碍。
The index of a subgroup of a group counts the number of cosets of that subgroup. A subgroup of finite index often shares structural properties with the group, and the existence of a subgroup of finite index with some particular property can therefore imply useful structural information for the overgroup. A developed theory of cosets in inverse semigroups exists, originally due to Schein: it is defined only for closed inverse subsemigroups, and the structural correspondences between an inverse semigroup and a closed inverse subsemigroup of finite index are weaker than in the group case. Neverth…
Let $G$ be a Garside group endowed with the generating set $\mathcal{S}$ of non-trivial simple elements, and let $H$ be a parabolic subgroup of $G$. We determine a transversal $T$ of $H$ in $G$ such that each $θ\in T$ is of minimal length in its right-coset, $H θ$, for the word length with respect to $\mathcal{S}$. We show that there exists a regular language $L$ on $\mathcal{S} \cup \mathcal{S}^{-1}$ and a bijection $\mathrm{ev} : L \to T$ satisfying $\mathrm{lg} (U) = \mathrm{lg}_\mathcal{S}( \mathrm{ev}(U))$ for all $U \in L$. From this we deduce that the coset growth series of $H$ in $G$ i…
A group and the completion of its coset semigroup
本文研究了群 $G$ 的非平凡子群右陪集逆子半群 ${\cal K}_1(G)$ 的完备化单位群，给出了极小子群相互置换的周期群的刻画与表示，并证明了除特殊 $p$-群外群与其陪集半群相互唯一决定，推广了 Schein 1973 年的结果。
Let $G$ be a finite non-solvable group. We prove that there exists a proper subgroup $A$ of $G$ such that $G$ is the product of three conjugates of $A$, thus replacing an earlier upper bound of $36$ with the smallest possible value. The proof relies on an equivalent formulation in terms of double cosets, and uses the following theorem which is of independent interest and wider scope: Any group $G$ with a $BN$-pair and a finite Weyl group $W$ satisfies $G=\left( Bn_{0}B\right) ^{2}=BB^{n_{0}}B$ where $n_{0}$ is any preimage of the longest element of $W$. The proof of the last theorem is formula…
This paper presents a solution of the polycirculant conjecture which states that every vertex-transitive graph G has an automorphism that permutes the vertices in cycles of the same length. This is done by identifying vertex-transitive graphs as coset graphs. For a coset graph H, an equivalence relation $\sim$ is defined on the vertices of cosets with classes as double cosets of the stabiliser and any other proper subgroup A' of a transitive group A of G. Induced left translations of elements of the subgroup A' are semi-regular since they preserve these double cosets and acts regularly on each…
Let $F$ be a finitely generated non-abelian free group and $Q$ a finite quotient. Denote by $L_Q$ the language obtained by adding unary predicates $P_q$, $q\in Q$ to the language of groups. Using a slight generalization of some of the techniques involved in Zlil Sela's solution to Tarskiś problem on the elementary theory of non-abelian free groups, we provide a few basic results on the validity of first order entences in the $L_Q$-expansion of $F$ in which every $P_q$ is interpreted as the preimage of $q$ in $F$. In particular we prove an analogous result to Sela's generalization of Merzlyakov…
Conjugacy Distinguished Cosets in Hyperbolic $3$-Manifold Groups
证明有限体积双曲3-流形群中，任何abelian子群的陪集是共轭区分的，并将此结果推广到loxodromic子群陪集与极大抛物子群类的共轭区分性。
Let $W$ be a finite coxeter group, let $W_I$, $W_J$ be standard parabolic subgroups, let $u$, $v$ be minimal double cosets representatives of double cosets in $W_I W / W_J$ and let $u'$, $v'$ be the maximal representatives of those same two cosets. We show that $u < v$ if and only if $u' < v'$ for the Bruhat order.
Retraction to a parabolic subgroup and applications
本文利用 Artin 群到标准抛物子群的左、右收缩，研究抛物子群的极小性、交、双陪集及共轭类等结构问题并获得新结论。
Character formula for conjugacy classes in a coset
本文研究了有限群中共轭类与不可约特征标之间的关系，特别探讨了在循环群情况下陪集中的共轭类数量与特征标扩展的关系。
We show that if $H$ is a quasiconvex subgroup of infinite index in a non-elementary hyperbolic group $G$ then the Schreier coset graph $X$ for $G$ relative to $H$ is non-amenable (that is, $X$ has positive Cheeger constant). We present some corollaries regading the Martin boundary and Martin compactification of $X$ and the co-growth of $H$ in $G$.
We show that the order complex of the poset of all cosets of all proper subgroups of a finite group $G$ is never $\mathbb{F}_{2}$-acyclic and therefore never contractible. This settles a question of K. S. Brown.
We show that if H is a quasiconvex subgroup of a hyperbolic group G then the relative Cayley graph Y (also known as the Schreier coset graph) for G/H is Gromov-hyperbolic. We also observe that in this situation if G is torsion-free and non-elementary and H has infinite index in G then the simple random walk on Y is transient.
Stallings folding theory is modified, using double coset representatives, and to applied to the study of subgroups of amalgamated products of finite rank free groups. As a first application the subgroup membership problem for such groups is shown to be decidable. An algorithm for this problem is constructed and its complexity is analysed. Groups in this class are also shown to possess the Howson property.
In this article we generalize the theory of subgroup graphs of subgroups of free groups to finite index subgroups $H$ of finitely generated groups $G$. We study and prove various properties of $H$ in relation to its subgroup graph $Γ(H)$. For a finitely generated group $G$ we consider the poset $P_{\text{fi}}(G)$ of all right cosets of all proper finite index subgroups of $G$. We use the theory of subgroup graphs to prove that for many finitely generated infinite groups the order complex $ΔP_{\text{fi}} (G)$ and the corresponding nerve complex are contractible.
On double coset separability and the Wilson-Zalesskii property
本文证明了每个双陪集可分群都享有Wilson-Zalesskii性质，并构造了一个不具备该性质且非双陪集可分的LERF群实例。
This is the author's second paper treating the double coset problem for classical groups. Let $G$ be an algebraic group over an algebraically closed field $K$. The double coset problem consists of classifying the pairs $H,J$ of closed connected subgroups of $G$ with finitely many $(H,J)$-double cosets in $G$. The critical setup occurs when one of $H,J$, say $H$, is reductive, and $J$ is a parabolic subgroup. Assume that $G$ is a classical group, $H$ is simple and $J$ is a maximal parabolic $P_k$, the stabilizer of a totally singular $k$-space. Then most candidates have $k=1$ or $k=2$. The case…
Group cosets with all elements of equal order
本文推广Camina对概念定义了有限群的等阶对，研究其性质，发现等阶对在多方面与Camina对相似但存在重要差异。
Let $G$ be a linear algebraic group defined over an algebraically closed field. The double coset question addressed in this paper is the following: Given closed subgroups $X$ and $P$, is the double coset collection $X\backslash G/P$ finite or infinite? We limit ourselves to the case where $X$ is maximal rank and reductive and $P$ parabolic. This paper presents a criterion for infiniteness which involves only dimensions of centralizers of semisimple elements. This result is then applied to finish the classification of those $X$ which are spherical. Finally, excluding a case in $F_4$, we show th…
Counting double cosets with application to generic 3-manifolds
本研究推广了双陪集增长理论至更广群类，证明 Morse 子群双陪集增长与轨道增长相当，并应用于证实 Maher 猜想：双曲三维流形在 Heegaard 分解流形中指数泛型。
We consider ordered tuples in finite groups generating nilpotent subgroups. Given an integer $q$ we consider the poset of nilpotent subgroups of class less than $q$ and its corresponding coset poset. These posets give rise to a family of finite Dirichlet series parametrized by the nilpotency class of the subgroups, which in turn reflect probabilistic and topological invariants determined by these subgroups. Connections of these series to filtrations of classifying spaces of a group are discussed.
The residual closure of a subgroup $H$ of a group $G$ is the intersection of all virtually normal subgroups of $G$ containing $H$. We show that if $G$ is generated by finitely many cosets of $H$ and if $H$ is commensurated, then the residual closure of $H$ in $G$ is virtually normal. This implies that separable commensurated subgroups of finitely generated groups are virtually normal. A stream of applications to separable subgroups, polycyclic groups, residually finite groups, groups acting on trees, lattices in products of trees and just-infinite groups then flows from this main result.
We consider a group $GL(\infty)$ and its parabolic subgroup $B$ corresponding to partition $\infty=\infty+m+\infty$. Denote by $P$ the kernel of the natural homomorphism $B\to GL(m)$. We show that the space of double cosets of $GL(\infty)$ by $P$ admits a natural structure of a semigroup. In fact this semigroup acts in subspaces of $P$-fixed vectors of some unitary representations of $GL(\infty)$ over finite field.
Topological concepts may be applied to any poset via the simplicial complex of finite chains. The coset poset C(G) of a finite group G (consisting of all cosets of all proper subgroups of G, ordered by inclusion) was introduced by Kenneth S. Brown in his study of the probabilistic zeta function P(G,s), and Brown's results motivate the study of the homotopy type of C(G). After a detailed introduction to basic simplicial topology and homotopy theory for finite posets (Chapter 2), we examine the connectivity of the coset poset (Chapter 3). Of particular interest is the fundamental group, and we p…
Elements of minimal length and Bruhat order on fixed point cosets of Coxeter groups
研究Coxeter群中固定点陪集上的Bruhat序，刻画最小长度元素并证明其与序极小性等价。
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