Dmitry Churikov, Ilia Ponomarenko
A permutation group $G\le\operatorname{Sym}(Ω)$ is said to be $2$-closed if no group $H$ such that $G<H\le\operatorname{Sym}(Ω)$ has the same orbits on $Ω\timesΩ$ as $G$. A simple and efficient inductive criterion for the $2$-closedness is established for abelian permutation groups with cyclic transitive constituents.
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We show that a finite permutation group containing a regular abelian self-normalizing subgroup is soluble.
Given a prime $p$, we construct a permutation group containing at least $p^{p-2}$ non-conjugated regular elementary abelian subgroups of order $p^3$. This gives the first example of a permutation group with exponentially many non-conjugated regular subgroups.
We study abelian-by-cyclic Moufang loops. We construct all split $3$-divisible abelian-by-cyclic Moufang loops from so-called Moufang permutations on abelian groups $(X,+)$, which are permutations that deviate from an automorphism of $(X,+)$ by an alternating biadditive mapping (satisfying certain properties). More generally, we obtain additional abelian-by-cyclic Moufang loops from so-called construction pairs. As an aside, we show that in the Moufang loops $Q$ obtained from a construction pair on $(X,+)$ the abelian normal subgroup $(X,+)$ induces an abelian congruence of $Q$ if and only if …
We present a construction of all finite indecomposable involutive solutions of the Yang-Baxter equation of multipermutational level at most 2 with abelian permutation group. As a consequence, we obtain a formula for the number of such solutions with a fixed number of elements. We also describe some properties of the automorphism groups in this case - in particular, we show they are regular abelian groups.
Let $G$ be a finite additive abelian group with exponent $n>1$, and let $a_1,\ldots,a_{n-1}\in G$. We show that there is a permutation $σ\in S_{n-1}$ such that all the elements $sa_{σ(s)}\ (s=1,\ldots,n-1)$ are nonzero if and only if $$\left|\left\{1\le s<n:\ \frac{n}{d}a_s\ne 0\right\}\right|\ge d-1\ \ \textrm{ for every positive divisor }\ d\ \textrm{ of }\ n.$$ When $G$ is the cyclic group $\mathbb Z/n\mathbb Z$, this confirms a conjecture of Z.-W. Sun.
Relatively closed subgroups of permutation groups with a cyclic regular normal subgroup
本文研究了具有 cyclic 正则正规子群的置换群中闭合子群的结构，通过分类最小非平凡一维仿射关联方案，扩展了最近的Muzychuk对一维仿射秩3图的分类结果。
We give a complete characterization of all indecomposable involutive solutions of the Yang-Baxter equation of multipermutation level~2. In the first step we present a construction of some family of such solutions and in the second step we prove that every indecomposable involutive solution of the Yang-Baxter equation with multipermutation level~2 is a homomorphic image of a solution previously constructed. Analyzing this epimorphism, we are able to obtain all such solutions up to isomorphism and enumerate these of small sizes.
We characterise the primitive 2-closed groups $G$ of rank at most four that are not the automorphism group of a graph or digraph and show that if the degree is at least 2402 then there are just two infinite families or $G\leqslant \mathrm{A}Γ\mathrm{L}_1(p^d)$, the 1-dimensional affine semilinear group. These are the first known examples of non-regular 2-closed groups that are not the automorphism group of a graph or digraph.
A finite group of order $n$ is said to have the distinct divisor property (DDP) if there exists a permutation $g_1,\ldots, g_n$ of its elements such that $g_i^{-1}g_{i+1} \neq g_j^{-1}g_{j+1}$ for all $1\leq i<j<n$. We show that an abelian group is DDP if and only if it has a unique element of order 2. We also describe a construction of DDP groups via group extensions by abelian groups and show that there exist infinitely many non abelian DDP groups.
The maximality of Abelian subgroups play a role in various parts of group theory. For example, Mycielski has extended a classical result of Lie groups and shown that a maximal Abelian subgroup of a compact connected group is connected and, furthermore, all the maximal Abelian subgroups are conjugate. For finite symmetric groups the question of the size of maximal Abelian subgroups has been examined by Burns and Goldsmith in 1989 and Winkler in 1993. We show that there is not much interest in generalizing this study to infinite symmetric groups; the cardinality of any maximal Abelian subgroup o…
Abelian groups are polynomially stable
该研究受Ornstein-Weiss拟平铺技术启发提出新方法，证明Abel群在对称群归一化汉明度量下关于置换具多项式稳定性，并给出其性质测试重述以提供高效置换性质测试器。
Finding cycle types in permutation groups with few generators
本文研究了置换群中寻找特定圈型置换的问题，证明了该问题对循环置换群可在对数空间求解，而对2生成的阿贝尔置换群及无不动点判定均为NP完全。
We explicitly determine all of the transitive groups of degree p-squared, p a prime, whose Sylow p-subgroup is not the wreath product of two cyclic groups of order p. Furthermore, we provide a general description of the transitive groups of degree p-squared whose Sylow p-subgroup is such a wreath product, and explicitly determine most of them. As applications, we solve the Cayley Isomorphism problem for Cayley objects of an abelian group of order p-squared, explicitly determine the full automorphism group of Cayley graphs of abelian groups of order p-squared, and find all nonnormal Cayley grap…
Let $N$ be a normal subgroup of a finite group $G$. For a faithful $N$-set $Δ$, applying the university embedding theorem one can construct a faithful $G$-set $Ω$. In this short note, it is proved that if the $2$-closure of $N$ in $Ω$ is equal to $N$, then the $2$-closure of $N$ in $Δ$ is also equal to $N$; in addition, it is proved that any abelian normal subgroup of a finite totally $2$-closed group is cyclic; finally, it is proved that if a finite nilpotent group is a direct of two nilpotent subgroups where the two factors have coprime orders and both of them are totally 2-closed then G is …
A group K is said to be a B-group if every permutation group containing K as a regular subgroup is either imprimitive or 2-transitive. In the second edition of his influential textbook on finite groups, Burnside published a proof that cyclic groups of composite prime-power degree are B-groups. Ten years later in 1921 he published a proof that every abelian group of composite degree is a B-group. Both proofs are character-theoretic and both have serious flaws. Indeed, the second result is false. In this note we explain these flaws and prove that every cyclic group of composite order is a B-grou…
Let X be a countably infinite set, Inj(X) the monoid of all injective endomaps of X, and Sym(X) the group of all permutations of X. We classify all submonoids of Inj(X) that are closed under conjugation by elements of Sym(X).
Studying solutions of the Yang-Baxter equation through skew braces, with an application to indecomposable involutive solutions with abelian permutation group
该研究建立 Yang-Baxter 方程集合论解与置换斜 brace 的性质关联，提出匹配的多重置换水平变体，关联轨道数与生成元、研究自同构群，还给出带阿贝尔置换群的有限不可分解对合解的描述及3级水平下的同构类数目。
On a huge family of non-schurian Schur rings
本文推广H·维兰特的例子，构造了偶秩初等阿贝尔群上的一大类非舒尔型舒尔环，解决了舒尔环是否为置换群传递模的问题。
Motivated by a problem from behavioral economics, we study subgroups of permutation groups that have a certain strong symmetry. Given a fixed permutation, consider the set of all permutations with disjoint inversion sets. The group is called non-nudgable, if the cardinality of this set always remains the same when replacing the initial permutation with its inverse. It is called nudgable otherwise. We show that all full permutation groups, standard dihedral groups, half of the alternating groups, and any abelian subgroup are non-nudgable. In the right probabilistic sense, it is thus quite likel…
We prove that all finitely generated free metabelian groups are permutation stable. This partially answers to the question asked by Levit and Lubotzky whether all finitely generated metabelian groups are permutation stable. Our proof extends the range of application of Levit-Lubotzky's method, which is used to show permutation stability of permutational wreath products of finitely generated abelian groups, to non-split and non-permutational metabelian groups.
Set-stabilizers in solvable permutation groups
本文证明了有限可解置换群中，模去可能平凡的正则初等交换3-子群后，必存在某个集合稳定子为2-群。
Let G,H be closed permutation groups on an infinite set X, with H a subgroup of G. It is shown that if G and H are orbit-equivalent, that is, have the same orbits on the collection of finite subsets of X, and G is primitive but not 2-transitive, then G=H.
We investigate finite non-Abelian simple groups $G$ for which the projective cover of the trivial module coincides with the permutation module on a subgroup and classify all cases unless $G$ is of Lie type in defining characteristic.
We determine a condition on the minimum Hamming weight of some special abelian group codes and, as a consequence of this result, we establish that any such code is, up to permutational equivalence, a subspace of the direct sum of $s$ copies of the repetition code of length $t$, for some suitable positive integers $s$ and $t$. Moreover, we provide a complete characterisation of permutation automorphisms of the linear code $C=\bigoplus_{i=1}^{s}Rep_{t}(\mathbb{F}_{q})$ and we establish that such a code is an abelian group code, for every pair of integers $s,t\geq1$. Finally, in a similar fashion…
Permutational wreath pullbacks and framed braid-type groups
本文引入置换 wreath 拉回结构，研究其中心和交换化，并探讨其作为经典 wreath 产品拉回的解释，同时建立有限生成阿贝尔群的条件，以继承 R_∞ 性质。
Locally compact abelian groups are classified in which the sum of any two closed subgroups is itself closed. This amounts to reproving and extending results by Yu.~N.~Mukhin from 1970. Namely we contribute a complete classification of all totally disconnected \lca\ groups with $X+Y$ closed for any closed subgroups $X$ and $Y$.
If f is a nonzero complex-valued function defined on a finite abelian group A and \hat f is its Fourier transform, then |Supp (f)||Supp {\hat f)| \ge |A|, where Supp (f) and Supp (\hat f) are the supports of f and \hat f. In this paper we generalize this known result in several directions. In particular, we prove an analogous inequality where the abelian group A is replaced by a transitive right G-set, where G is an arbitrary finite group. We obtain stronger inequalities when the G-set is primitive and we determine the primitive groups for which equality holds. We also explore connections betw…
There is a connection between permutation groups and permutation patterns: for any subgroup $G$ of the symmetric group $S_\ell$ and for any $n \geq \ell$, the set of $n$-permutations involving only members of $G$ as $\ell$-patterns is a subgroup of $S_n$. Making use of the monotone Galois connection induced by the pattern avoidance relation, we characterize the permutation groups that arise via pattern avoidance as automorphism groups of relations of a certain special form. We also investigate a related monotone Galois connection for permutation groups and describe its closed sets and kernels …
Let $G \leq \mathrm{Sym} (X)$ for a countable set $X$. Call a colouring of $X$ asymmetric, if the identity is the only element of $G$ which preserves all colours. The motion (also called minimal degree) of $G$ is the minimal number of elements moved by an element $g \in G \setminus\{\mathrm{id}\}$. We show that every locally compact, closed permutation group with infinite motion admits an asymmetric $2$-colouring. This generalises a recent result by Babai and confirms a conjecture by Imrich, Smith, Tucker, and Watkins from 2015.
Skew-product groups of finite abelian $p$-groups
本文研究有限阿贝尔 $p$-群的斜积群，将此前对有限初等阿贝尔 $p$-群的结果推广到更一般的有限阿贝尔 $p$-群情形。
In this thesis we study the following property of a finite group G: the minimal number n such that G embeds in Sn. We start with an explicit formula for the number n for abelian groups. Then, we study the behavior of this group property in respect to direct products. Finally, we define and explore the "compression ratio" of a finite group G which measures how much better the best embedding is relative to the embedding given by Cayley's theorem.
In arXiv:0910.1727 we find certain finite homomorphic images of Artin braid group into appropriate symmetric groups, which a posteriori are extensions of the symmetric group on n letters by an abelian group. The main theorem of this paper characterizes completely the extensions of this type that are split.
This paper is devoted to the proof of a structural theorem, concerning certain homomorphic images of Artin braid group on $n$ strands in finite symmetric groups. It is shown that any one of these permutation groups is an extension of the symmetric group on $n$ letters by an appropriate abelian group, and in "half" of the cases this extension splits.
In this paper we study the problem of classification of indecomposable solutions of the Yang-Baxter equation. Using a scheme proposed by Bachiller, Cedó, and Jespers, and recent advances in the classification of braces we classify all indecomposable solutions with some particular permutation groups. We do this for all groups of size $pq$, all abelian groups of size $p^2q$ and all dihedral groups of size $p^2q$.
In this paper, I show that if $p$ is an odd prime, and if $P$ is a finite $p$-group, then there exists an exact sequence of abelian groups $$0\to T(P)\to D(P)\to\lproj{P}\to H^1\big(\apdeux(P),\Z\big)^{(P)},$$ where $D(P)$ is the Dade group of $P$ and $T(P)$ is the subgroup of endo-trivial modules. Here $\lproj{P}$ is the group of sequences of compatible elements in the Dade groups $D\big(N_P(Q)/Q\big)$ for non trivial subgroups $Q$ of $P$. The poset $\apdeux(P)$ is the set of elementary abelian subgroups of rank at least 2 of $P$, ordered by inclusion. The group $H^1\big(\apdeux(P),\Z\big)^{(…
On endosplit $p$-permutation resolutions and Broué's conjecture for $p$-solvable groups
本文通过端分裂 $p$-置换分解的新刻画和 Galois 下降的约化，证明了 Kessar-Linckelmann 提出的 Broué 猜想细化版本对满足 $G = O_{p',p,p'}(G)$ 且具有交换 Sylow $p$-子群的某些块成立，这是 Harris-Linckelmann 验证所有 $p$-可解群的 Broué 猜想的关键约化步骤。
Bounding the order of complex linear groups and permutation groups with selected composition factors
受对称张量范畴中张量幂增长率问题启发，本文给出合成因子受限的复一般线性群有限子群阶的一般界，并拓展其背景与应用。
We classify the locally compact second-countable (l.c.s.c.) groups $A$ that are abelian and topologically characteristically simple. All such groups $A$ occur as the monolith of some soluble l.c.s.c. group $G$ of derived length at most $3$; with known exceptions (specifically, when $A$ is $\mathbb{Q}^n$ or its dual for some $n \in \mathbb{N}$), we can take $G$ to be compactly generated. This amounts to a classification of the possible isomorphism types of abelian chief factors of l.c.s.c. groups, which is of particular interest for the theory of compactly generated locally compact groups.
Let $k$ be a field of characteristic $p > 0$. For $G$ an elementary abelian $p$-group, there exist collections of permutation module such that if $C^*$ is any exact bounded complex whose terms are sums of copies of modules from the collection, then $C^*$ is contractible. A consequence is that if $G$ is any finite group whose Sylow $p$-subgroups are not cyclic or quaternion, and if $C^*$ is a bounded exact complex such that each $C^i$ is direct sum of one dimensional modules and projective modules, then $C^*$ is contractible.
It is shown that square free set theoretic involutive non-degenerate solutions of the Yang-Baxter equation whose associated permutation group (referred to as an involutive Yang-Baxter group) is abelian are retractable in the sense of Etingof, Schedler and Soloviev. This solves a problem of Gateva-Ivanova in the case of abelian IYB groups. It also implies that the corresponding finitely presented abelian-by-finite groups (called the structure groups) are poly-${\mathbb Z}$ groups. Secondly, an example of a solution with an abelian involutive Yang-Baxter group which is not a generalized twisted …
Here we show that a finite nilpotent group is 2-closed if and only if it is either cyclic or a direct product of a generalized quaternion group with a cyclic group of odd order.
Let G be a finite abelian group with |G|>1. Let a_1,...,a_k be k distinct elements of G and let b_1,...,b_k be (not necessarily distinct) elements of G, where k is a positive integer smaller than the least prime divisor of |G|. We show that there is a permutation $π$ on {1,...,k} such that a_1b_{π(1)},...,a_kb_{π(k)} are distinct, provided that any other prime divisor of |G| (if there is any) is greater than k!. This in particular confirms the Dasgupta-Karolyi-Serra-Szegedy conjecture for abelian p-groups. We also pose a new conjecture involving determinants and characters, and show that its v…
A-nuclei (groups of regular permutations) of a quasigroup are studied. A quasigroup is A-nuclear if and only if it is group isotope. Any quasigroup with permutation medial or paramedial identity is an abelian group isotope. Definition of A-center of a quasigroup is given. A quasigroup is A-central if and only if it is abelian group isotope. If a quasigroup is central in Belyavskaya-Smith sense, then it is A-central. Conditions when A-nucleus define normal congruence of a quasigroup are established, conditions normality of nuclei of some inverse quasigroups are given. Notice, definition of A-nu…
The group A_{m} of automophisms of a one-rooted m-ary tree admits a diagonal monomorphism which we denote by x. Let A be an abelian state-closed (or self-similar) subgroup of A_{m}. We prove that the combined diagonal and tree-topological closure A* of A is additively a finitely presented Z_m [[x]]-module where Z_m is the ring of m-adic integers. Moreover, if A* is torsion-free then it is a finitely generated pro-m group. The group A splits over its torsion subgroup. We study in detail the case where A* corresponds to a cyclic Z_m[[x]]-module and when m is a prime number, we show A* to be conj…
For a set $Ω$ an unordered relation on $Ω$ is a family R of subsets of $Ω.$ If R is such a relation we let G(R) be the group of all permutations on $Ω$ that preserves R, that is g belongs to G(R) if and only if x in R implies x^{g}\in R. We are interested in permutation groups which can be represented as G=G(R) for a suitable unordered relation R on $Ω.$ When this is the case, we say that G is defined by the relation R, or that G is a relation group. We prove that a primitive permutation group different from the Alternating Group and of degree bigger or equal to 11 is a relation groups. The sa…
On the relation of character codegrees and the minimal faithful quasi-permutation representation degree of $p$-groups
本文研究有限$p$-群的最小忠实拟置换表示次数$c(G)$与不可约特征标余次数的关系，建立了奇数阶非阿贝尔$p$-群中$c(G)$与特征标余次数非负有理和的等式，并探讨多类非阿贝尔$p$-群的对应关系。
The Brauer relations of a finite group $G$ are virtual differences of non-isomorphic $G$-sets $X-Y$ which induce isomorphic permutation $G$-representations $\mathbb Q[X]\simeq\mathbb Q[Y]$ over the rationals. These relations have been classified by Tornehave-Bouc and Bartel-Dokchitser. Motivated by stable homotopy theory, a relative version of Brauer relations for $(G,C_p)$-bisets which are $C_p$-free have been classified by Kahn in case $G$ is an elementary Abelian $p$-group. In this paper we extend Kahn's classification to the case when $G$ is a finite Abelian $p$-group.
Complexity of Minimal Faithful Permutation Degree for Fitting-free Groups
本文研究无 Abel 正规子群的群的最小忠实置换度计算复杂度，证明该问题在以置换群商给出时属于 P，并在置换群设定下获得 NC 上界，改进了 Das 和 Thakkar 的先前工作。
Subsets of abelian groups closed under addition or subtraction
本文研究了交换群中闭合于加法或减法的子集，发现其结构为至多两个元素或无限集{rk}，并探讨了非交换群的类似问题。
The minimal faithful permutation degree $μ(G)$ of a finite group $G$ is the least nonnegative integer $n$ such that $G$ embeds in the symmetric group $\Sym(n)$. We prove that if $H$ is a group then $μ(G)=μ(G\times H)$ for some group $G$ then $H$ embeds in $A\times Q^k$ for some abelian group of odd order, some generalised quaternion $2$-group and some nonnegative integer $k$. As a consequence, $μ(G^{n+1})=μ(G^n)$ for some nonnegative integer $n$ if and only if $G$ is trivial.
A permutation group is innately transitive if it has a transitive minimal normal subgroup, and this subgroup is called a plinth. In this paper we study three special types of inclusions of innately transitive permutation groups in wreath products in product action. This is achieved by studying the natural Cartesian decomposition of the underlying set that correspond to the product action of a wreath product. Previously we identified six classes of Cartesian decompositions that can be acted upon transitively by an innately transitive group with a non-abelian plinth. The inclusions studied in th…
Using the diagrammatic approach to integrals over Gaussian random matrices, we find a representation for polynomial Lie group integrals as infinite sums over certain maps on surfaces. The maps involved satisfy a specific condition: they have some marked vertices, and no closed walks that avoid these vertices. We also formulate our results in terms of permutations, arriving at new kinds of factorization problems.
We study the base sizes of finite quasiprimitive permutation groups of twisted wreath type, which are precisely the finite permutation groups with a unique minimal normal subgroup that is also non-abelian, non-simple and regular. Every permutation group of twisted wreath type is permutation isomorphic to a twisted wreath product $G=T^k{:}P$ acting on its base group $Ω=T^k$, where $T$ is some non-abelian simple group and $P$ is some group acting transitively on $\boldsymbol{k}=\{1,\ldots,k\}$ with $k\geq 2$. We prove that if $G$ is primitive on $Ω$ and $P$ is quasiprimitive on $\boldsymbol{k}$,…
W.H.~Mills has determined, for a finitely generated abelian group $G$, the regular subgroups $N \cong G$ of $S(G)$, the group of permutations on the set $G$, which have the same holomorph of $G$, that is, such that $N_{S(G)}(N) = N_{S(G)}(ρ(G))$, where $ρ$ is the (right) regular representation. We give an alternative approach to Mills' result, which relies on a characterization of the regular subgroups of $N_{S(G)}(ρ(G))$ in terms of commutative ring structures on $G$. We are led to solve, for the case of a finitely generated abelian group $G$, the following problem: given an abelian group $(G…
We study groups that can be defined as Polish, pro-countable groups, as non-archimedean groups with an invariant metric or as quasi-countable groups, i.e., closed subdirect products of countable, discrete groups, endowed with the product topology. We show, among other results, that for every non-locally compact, abelian quasi-countable group G there exists a closed subgroup L of G, and a closed, non-locally compact subgroup K of G/L which is a direct product of discrete, countable groups. As an application we prove that for every abelian Polish group G of the form H/L, where H,L are closed sub…
The complexity of a geodesic language has connections to algebraic properties of the group. Gilman, Hermiller, Holt, and Rees show that a finitely generated group is virtually free if and only if its geodesic language is locally excluding for some finite inverse-closed generating set. The existence of such a correspondence and the result of Hermiller, Holt, and Rees that finitely generated abelian groups have piecewise excluding geodesic language for all finite inverse-closed generating sets motivated our work. We show that a finitely generated group with piecewise excluding geodesic language …
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 $g$, …
Hyperplanes in abelian groups and twisted signatures
本文研究有限循环群乘积之间保持坐标超平面并集的同构条件，证明此类同构在因子阶大于2时必为置换意义下的对角映射，并利用扭曲签名构造了一族可由第一同调对角矩阵刻画唯一分解性的紧4-流形。
We describe all closed permutation groups which act on the set of vectors of a countable vector space $V$ over a prime field of odd order and which contain all automorphisms of $V$. In particular, we prove that their number is finite. These groups correspond, up to first-order interdefinability, precisely to all structures with a first-order definition in $V$.
We exhibit a closed aspherical 5-manifold of nonpositive curvature that fibers over a circle whose fundamental group is hyperbolic relative to abelian subgroups such that the fiber is a closed aspherical 4-manifold whose fundamental group is not hyperbolic relative to abelian subgroups.
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