Wreath Product Research Articles

The degree of commutativity of wreath products with infinite cyclic top group

The degree of commutativity of a finite group is the probability that two uniformly and randomly chosen elements commute. This notion extends naturally to finitely generated groups G: the degree of commutativity dcS(G)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${{\\,\ extrm{dc}\\,}}_S(G)$$\\end{document}, with respect to a given finite generating set S, results from considering the fractions of commuting pairs of elements in increasing balls around 1G\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$1_G$$\\end{document} in the Cayley graph . We focus on restricted wreath products of the form G=H≀⟨t⟩\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$G = H \\hspace{1.111pt}{\\wr }\\hspace{1.111pt}\\langle \\hspace{1.111pt}t \\rangle $$\\end{document}, where H≠1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$H \ e 1$$\\end{document} is finitely generated and the top group ⟨t⟩\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\langle \\hspace{1.111pt}t \\rangle $$\\end{document} is infinite cyclic. In accordance with a more general conjecture, we show that dcS(G)=0\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${{\\,\ extrm{dc}\\,}}_S(G) = 0$$\\end{document} for such groups G, regardless of the choice of S. This extends results of Cox who considered lamplighter groups with respect to certain kinds of generating sets. We also derive a generalisation of Cox’s main auxiliary result: in ‘reasonably large’ homomorphic images of wreath products G as above, the image of the base group has density zero, with respect to certain types of generating sets.

Topological actions of wreath products

Topological actions of wreath products

Wreath products of groups acting with bounded orbits

Wreath products of groups acting with bounded orbits

Determination of the Conjugacy Classes and Character Table of the Full Non-Rigid Group of Hexachlorocyclopropane Chemical Compound Via Wreath Product of Pair of Permutation Groups

The use of full non-rigid (f.NRG) molecules group theory to study the internal dynamics of molecular structures of chemical compounds is trending in the research space. In this paper, we use computational method to compute the group elements and group table of the Hexachlorocyclopropane molecular (algebraic) structure and thereafter determine the order and conjugacy classes of the group and finally the corresponding symmetry for each permutation group. We considered the point group of the compound which turns out to be isomorphic to the Wreath Products C3wrC2, where Cn denotes a cyclic group of order n. We found that the group has order 18 and 9 conjugacy classes. Investigating the solubility of the components of Hexachlorocyclopropane chemical compound can be considered for further study.

Wreath Products and Projective System of non-Schurian Association Schemes

A wreath product is a method to construct an association scheme from two association schemes. We determine the automorphism group of a wreath product. We show a known result that a wreath product is Schurian if and only if both components are Schurian, which yields large families of non-Schurian association schemes and non-Schurian $S$-rings. We also study iterated wreath products. Kernel schemes by Martin and Stinson are shown to be iterated wreath products of class-one association schemes. The iterated wreath products give examples of projective systems of non-Schurian association schemes, with an explicit description of primitive idempotents.

Distinguishing labelling of sets under the wreath product action of $$\overrightarrow{A_{n}}$$

Distinguishing labelling of sets under the wreath product action of $$\overrightarrow{A_{n}}$$

Almost all wreath product character values are divisible by given primes

Almost all wreath product character values are divisible by given primes

Reidemeister Classes, Wreath Products, and Solvability

Reidemeister Classes, Wreath Products, and Solvability

Amenable wreath products with non almost finite actions of mean dimension zero

Almost finiteness was introduced in the seminal work of Kerr [J. Eur. Math. Soc. (JEMS) 22 (2020), pp. 3697–3745] as an dynamical analogue of Z \mathcal {Z} -stability in the Toms-Winter conjecture. In this article, we provide the first examples of minimal, topologically free actions of amenable groups that have mean dimension zero but are not almost finite. More precisely, we prove that there exists an infinite family of amenable wreath products that admit topologically free, minimal profinite actions on the Cantor space which fail to be almost finite. Furthermore, these actions have dynamical comparison. This intriguing new phenomenon shows that Kerr’s dynamical analogue of Toms-Winter conjecture fails for minimal, topologically free actions of amenable groups. The notion of allosteric group holds a significant position in our study. A group is allosteric if it admits a minimal action on a compact space with an invariant ergodic probability measure that is topologically free but not essentially free. We study allostery of wreath products and provide the first examples of allosteric amenable groups.

The Wreath Product of Powerful p-Groups

This study provides a scholarly examination of fundamental concepts within the field of group theory, specifically focusing on topics such as the wreath product and powerful p-groups. We examine the characteristics pertaining to the structure of the wreath product of cyclic p-groups, with a specific focus on the groups that are powerfully embedded within it. The primary discovery pertains to the construction of the powerful wreath product and the quasi-powerful wreath product. In this study, we establish that subgroups are powerful within the wreath product, specifically focusing on p-groups. The aforementioned outcome is derived from the assumption that p is a prime number and W is the standard wreath product of two nontrivial cyclic p-groups, denoted as G and H.

Bounding conjugacy depth functions for wreath products of finitely generated abelian groups

In this article, we study the asymptotic behaviour of conjugacy separability for wreath products of abelian groups. We fully characterise the asymptotic class in the case of lamplighter groups and give exponential upper and lower bounds for generalised lamplighter groups. In the case where the base group is infinite, we give superexponential lower and upper bounds. We apply our results to obtain lower bounds for conjugacy depth functions of various wreath products of groups where the acting group is not abelian.

Multiple partition structures and harmonic functions on branching graphs

We introduce and study multiple partition structures which are sequences of probability measures on families of Young diagrams subjected to a consistency condition. The multiple partition structures are generalizations of Kingman's partition structures, and are motivated by a problem of population genetics. They are related to harmonic functions and coherent systems of probability measures on a certain branching graph. The vertices of this graph are multiple Young diagrams (or multiple partitions), and the edges depend on the Jack parameter. If the value of the Jack parameter is equal to one the branching graph under considerations reflects the branching rule for the irreducible representations of the wreath product of a finite group with the symmetric group. If the value of the Jack parameter is zero then the coherent systems of probability measures are precisely the multiple partition structures. We establish a bijective correspondence between the set of harmonic functions on the graph and probability measures on the generalized Thoma set. The correspondence is determined by a canonical integral representation of harmonic functions. As a consequence we obtain a representation theorem for multiple partition structures.We give an example of a multiple partition structure which is expected to be relevant for a model of population genetics for the genetic variation of a sample of gametes from a large population. Namely, we construct a probability measure on the wreath product of a finite group with the symmetric group. If the finite group contains the identity element only then it coincides with the well-known Ewens probability measure on the symmetric group. The constructed probability measure defines a multiple partition structure which is a generalization of the Ewens partition structure studied by Kingman. We show that this multiple partition structure can be represented in terms of a multiple analogue of the Poisson-Dirichlet distribution called the multiple Poisson-Dirichlet distribution in the paper.

Pre-primitive permutation groups

A transitive permutation group G on a finite set Ω is said to be pre-primitive if every G-invariant partition of Ω is the orbit partition of a subgroup of G. It follows that pre-primitivity and quasiprimitivity are logically independent (there are groups satisfying one but not the other) and their conjunction is equivalent to primitivity. Indeed, part of the motivation for studying pre-primitivity is to investigate the gap between primitivity and quasiprimitivity. We investigate the pre-primitivity of various classes of transitive groups including groups with regular normal subgroups, direct and wreath products, and diagonal groups. In the course of this investigation, we describe all G-invariant partitions for various classes of permutation groups G. We also look briefly at conditions similarly related to other pairs of conditions, including transitivity and quasiprimitivity, k-homogeneity and k-transitivity, and primitivity and synchronization.

Dead ends on wreath products and lamplighter groups

For any finite group A and any finitely generated group B, we prove that the corresponding lamplighter group [Formula: see text] admits a standard generating set with unbounded depth, and that if B is abelian then the above is true for every standard generating set. This generalizes the case where [Formula: see text] together with its cyclic generator [S. Cleary and J. Taback, Dead end words in lamplighter groups and other wreath products, Q. J. Math. 56(2) (2005) 165–178]. When [Formula: see text] is the free product of two finite groups H and K, we characterize which standard generators of the associated lamplighter group have unbounded depth in terms of a geometrical constant related to the Cayley graphs of H and K. In particular, we find differences with the one-dimensional case: the lamplighter group over the free product of two sufficiently large finite cyclic groups has uniformly bounded depth with respect to some standard generating set.

Multiparameter colored partition category and the product of the reduced Kronecker coefficients

We introduce and study a multiparameter colored partition category CPar(x) by extending the construction of the partition category, over an algebraically closed field k of characteristic zero and for a multiparameter x∈kr. The morphism spaces in CPar(x) have bases in terms of partition diagrams whose parts are colored by elements of the multiplicative cyclic group Cr. We show that the endomorphism spaces of CPar(x) and additive Karoubi envelope of CPar(x) are generically semisimple. The category CPar(x) is rigid symmetric strict monoidal and we give a presentation of CPar(x) as a monoidal category. The path algebra of CPar(x) admits a triangular decomposition with Cartan subalgebra being equal to the direct sum of the group algebras of complex reflection groups G(r,n). We compute the structure constants for the classes of simple modules in the split Grothendieck ring of the category of modules over the path algebra of the downward partition subcategory of CPar(x) in two ways. Among other things, this gives a formula for the product of the reduced Kronecker coefficients in terms of the Littlewood–Richardson coefficients for G(r,n) and certain Kronecker coefficients for the wreath product (Cr×Cr)≀Sn. For r=1, this formula reduces to a formula for the reduced Kronecker coefficients given by Littlewood. We also give two analogues of the Robinson–Schensted correspondence for colored partition diagrams and, as an application, we classify the equivalence classes of Green's left, right and two-sided relations for the colored partition monoid in terms of these correspondences.

Derangements in wreath products of permutation groups

Given a finite group G acting on a set X let δk(G,X)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\delta _k(G,X)$$\\end{document} denote the proportion of elements in G that have exactly k fixed points in X. Let Sn\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathcal {S}_n$$\\end{document} denote the symmetric group acting on [n]={1,2,⋯,n}\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$[n]=\\{1,2,\\dots ,n\\}$$\\end{document}. For A⩽Sm\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$A\\leqslant \\mathcal {S}_m$$\\end{document} and B⩽Sn\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$B\\leqslant \\mathcal {S}_n$$\\end{document}, the permutational wreath product A≀B\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$A\\wr B$$\\end{document} has two natural actions and we give formulas for both, δk(A≀B,[m]×[n])\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\delta _k(A\\wr B,[m]{\ imes }[n])$$\\end{document} and δk(A≀B,[m][n])\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\delta _k(A\\wr B,[m]^{[n]})$$\\end{document}. We prove that for k=0\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$k=0$$\\end{document} the values of these proportions are dense in the intervals [δ0(B,[n]),1]\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$[\\delta _0(B,[n]),1]$$\\end{document} and [δ0(A,[m]),1]\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$[\\delta _0(A,[m]),1]$$\\end{document}. Among further results, we provide estimates for δ0(G,[m][n])\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\delta _0(G,[m]^{[n]})$$\\end{document} for subgroups G⩽Sm≀Sn\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$G\\leqslant \\mathcal {S}_m\\wr \\mathcal {S}_n$$\\end{document} containing Am[n]\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathcal {A}_m^{[n]}$$\\end{document}.

On the normalizer of an iterated wreath product

Given a group G and n ≥ 0 , let W(G, n) be the associated iterated wreath product—unrestricted when G is infinite—viewed as a permutation group on Gn . We prove that the normalizer of W(G, n) in the symmetric group S ( G n ) is equal to M n ⋉ W ( G , n ) , where Mn is isomorphic to Aut ( G ) n . The action of Aut ( G ) n on W(G, n) is recursively described. Communicated by Mark Lewis

Measure equivalence rigidity via s-malleable deformations

We single out a large class of groups ${\rm {\boldsymbol {\mathscr {M}}}}$ for which the following unique prime factorization result holds: if $\Gamma _1,\ldots,\Gamma _n\in {\rm {\boldsymbol {\mathscr {M}}}}$ and $\Gamma _1\times \cdots \times \Gamma _n$ is measure equivalent to a product $\Lambda _1\times \cdots \times \Lambda _m$ of infinite icc groups, then $n \ge m$ , and if $n = m$ , then, after permutation of the indices, $\Gamma _i$ is measure equivalent to $\Lambda _i$ , for all $1\leq i\leq n$ . This provides an analogue of Monod and Shalom's theorem [Orbit equivalence rigidity and bounded cohomology, Ann. of Math. 164 (2006), 825–878] for groups that belong to ${\rm {\boldsymbol {\mathscr {M}}}}$ . Class ${\rm {\boldsymbol {\mathscr {M}}}}$ is constructed using groups whose von Neumann algebras admit an s-malleable deformation in the sense of Sorin Popa and it contains all icc non-amenable groups $\Gamma$ for which either (i) $\Gamma$ is an arbitrary wreath product group with amenable base or (ii) $\Gamma$ admits an unbounded 1-cocycle into its left regular representation. Consequently, we derive several orbit equivalence rigidity results for actions of product groups that belong to ${\rm {\boldsymbol {\mathscr {M}}}}$ . Finally, for groups $\Gamma$ satisfying condition (ii), we show that all embeddings of group von Neumann algebras of non-amenable inner amenable groups into $L(\Gamma )$ are ‘rigid’. In particular, we provide an alternative solution to a question of Popa that was recently answered by Ding, Kunnawalkam Elayavalli, and Peterson [Properly Proximal von Neumann Algebras, Preprint (2022), arXiv:2204.00517].

Nil algebras, Lie algebras and wreath products with intermediate and oscillating growth

We construct finitely generated nil algebras with prescribed growth rates. In particular, any increasing submultiplicative function is realized as the growth function of a nil algebra up to a polynomial error term and an arbitrarily slow distortion.We then move on to examples of nil algebras and domains with strongly oscillating growth functions and construct primitive algebras for which the Gel'fand-Kirillov dimension is strictly sub-additive with respect to tensor products, thus answering a question from [16,17].

Lengths of cycles in generalized pancake graphs

In this paper, we consider the lengths of cycles that can be embedded on the edges of the generalized pancake graphs which are the Cayley graph of the generalized symmetric group S(m,n), generated by prefix reversals. The generalized symmetric group S(m,n) is the wreath product of the cyclic group of order m and the symmetric group of order n!. Our main focus is the underlying undirected graphs, denoted by Pm(n). In the cases when the cyclic group has one or two elements, these graphs are isomorphic to the pancake graphs and burnt pancake graphs, respectively. We prove that when the cyclic group has three elements, P3(n) has cycles of all possible lengths, thus resembling a similar property of pancake graphs and burnt pancake graphs. Moreover, P4(n) has all the even-length cycles. We utilize these results as base cases and show that if m>2 is even, Pm(n) has all cycles of even length starting from its girth to a Hamiltonian cycle. Moreover, when m>2 is odd, Pm(n) has cycles of all lengths starting from its girth to a Hamiltonian cycle. We furthermore show that the girth of Pm(n) is min⁡{m,6} if m≥3, thus complementing the known results for m=1,2.