Transfinite hypercentral iterated wreath product of integral domains

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.

Let G be a group which has a finite number r_n(G) of irreducible linear representations in \mathrm{GL}_n(\mathbb C) for all n \geq 1 . Let \zeta(G,s) = \sum_{n = 1}^\infty r_n(G)n^{−s} be its representation zeta function. First, in case G = H \wr_X Q is a permutational wreath product with respect to a permutation group Q on a finite set X , we establish a formula for \zeta(G,s) in terms of the zeta functions of H and of subgroups of Q , and of the Möbius function associated to the lattice \Pi_Q(X) of partitions of X in orbits under subgroups of Q . Then we consider groups W(Q,k) = (\cdots (Q \wr_X Q) \wr_X Q ···) \wr_X Q which are iterated wreath products (with k factors Q ), and several related infinite groups W(Q) , including the profinite group \lim_{\leftarrow k}W(Q,k) , a locally finite group \lim_kW(Q,k) , and several finitely generated dense subgroups of \lim_{\leftarrow k}W(Q,k) . Under convenient hypotheses (in particular Q should be perfect), we show that r_n(W(Q)) < \infty for all n \geq 1 , and we establish that the Dirichlet series \zeta(W(Q),s) has a finite and positive abscissa of convergence \sigma_0 = \sigma_0(W(Q)) . Moreover, the function \zeta(W(Q),s) satisfies a remarkable functional equation involving \zeta(W(Q),es) for e \in \{1,\dots,d\} , where d = |X| . As a consequence of this, we exhibit some properties of the function, in particular that \zeta(W(Q),s) has a root-type singularity at \sigma_0 , with a finite value at \sigma_0 and a Puiseux expansion around \sigma_0 . We finally report some numerical computations for Q = A_5 and Q = \mathrm{PGL}_3(\mathbb F_2) .

We show that the probability of generating an iterated wreath product of non-abelian finite simple groups converges to 1 as the order of the first simple group tends to infinity provided the wreath products are constructed with transitive and faithful actions. This has the consequence that the profinite group which is the inverse limit of these iterated wreath products is positively finitely generated.

We study the neighborhoods of a typical point $$Z_n$$ visited at n-th step of a random walk, determined by the condition that the transition probabilities stay close to $$\mu ^{*n}(Z_n)$$ . If such neighborhood contains a ball of radius $$C \sqrt{n}$$ , we say that the random walk has almost invariant transition probabilities. We prove that simple random walks on wreath products of $$\mathbb {Z}$$ with finite groups have almost invariant distributions. A weaker version of almost invariance implies a necessary condition of Ozawa’s criterion for the property $$H_\mathrm{FD}$$ . We define and study the radius of almost invariance. We estimate this radius for random walks on iterated wreath products and show this radius can be asymptotically strictly smaller than n / L(n), where L(n) denotes the drift function of the random walk. We show that the radius of individual almost invariance of a simple random walk on the wreath product of $$\mathbb {Z}^2$$ with a finite group is asymptotically strictly larger than n / L(n). Finally, we show the existence of groups such that the radius of almost invariance is smaller than a given function, but remains unbounded. We also discuss possible limiting distribution of ratios of transition probabilities on non almost invariant scales.

Let (n 1, n 2,…) be an infinite sequence of positive integers greater than 1. We prove that the topological rank of the corresponding infinitely iterated wreath product …≀C n 2 ≀C n 1 of finite cyclic groups is equal to k + 1, where k denotes the topological rank of the direct product C n 2 × C n 3 × … of finite cyclic groups (this rank may be of finite or infinite cardinality, that is, k is a finite number or k = ∞). We construct the minimal set generating topologically the above wreath product as a set of the so-called cyclic automorphisms of a spherically homogeneous rooted tree.

Let CAn=C[S2≀S2≀⋯≀S2] be the group algebra of an n-step iterated wreath product. We prove some structural properties of An such as their centers, centralizers, and right and double cosets. We apply these results to explicitly write down the Mackey theorem for groups An and give a partial description of the natural transformations between induction and restriction functors on the representations of the iterated wreath product tower by computing certain hom spaces of the category of ⨁m≥0(Am,An)−bimodules. A complete description of the category is an open problem.

‎‎In this paper we prove that the Maschke property holds for coprime actions on some important classes of $p$-groups like‎: ‎metacyclic $p$-groups‎, ‎$p$-groups of $p$-rank two for $p>3$ and some weaker property holds in the case of regular $p$-groups‎. ‎The main focus will be the case of coprime actions on the iterated wreath product $P_n$ of cyclic groups of order $p$‎, ‎i.e‎. ‎on Sylow $p$-subgroups of the symmetric groups $S_{p^n}$‎, ‎where we also prove that a stronger form of the Maschke property holds‎. ‎These results contribute to a future possible classification of all $p$-groups with the Maschke property‎. ‎We apply these results to describe which normal partition subgroups of $P_n$ have a complement‎. ‎In the end we also describe abelian subgroups of $P_n$ of largest size‎.

We consider faithful finitary linear representations of (generalized) wreath products A wrΩH of groups A by H over (potentially) infinite-dimensional vector spaces, having previously considered completely reducible such representations in an earlier paper. The simpler the structure of A the more complex, it seems, these representations can become. If A has no non-trivial abelian normal subgroups, the conditions we present are both necessary and sufficient. They imply, for example, that for such an A, if there exists such a representation of the standard wreath product A wr H of infinite dimension, then there already exists one of finite dimension.

Extremal and maximal normal abelian subgroups of a maximal unipotent subgroup in groups of Lie type

The isomorphism problem for groups given by their multiplication tables (GPI) has long been known to be solvable in n <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">O(log n)</sup> time, but only recently has there been significant progress towards polynomial time. For example, Babai et al. (ICALP 2012) gave a polynomial-time algorithm for groups with no abelian normal subgroups. Thus, at present it is crucial to understand groups with abelian normal subgroups to develop n <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">o(log n)</sup> -time algorithms. Towards this goal we advocate a strategy via the extension theory of groups, which describes how a normal subgroup N is related to G/N via G. This strategy "splits" GPI into two sub problems: one regarding group actions on other groups, and one regarding group cohomology. The solution of these problems is essentially necessary and sufficient to solve GPI. Most previous works naturally align with this strategy, and it thus helps explain in a unified way the recent polynomial-time algorithms for other group classes. In particular, most prior results in the multiplication table model focus on the group action aspect, despite the general necessity of cohomology, for example for p-groups of class 2-believed to be the hardest case of GPI. To make progress on the group cohomology aspect of GPI, we consider central-radical groups, proposed in Babai et al. (SODA 2011): the class of groups such that G mod its center has no abelian normal subgroups. Recall that Babai et al. (ICALP 2012) consider the class of groups G such that G itself has no abelian normal subgroups. Following the above strategy, we solve GPI in n <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">O(log log n)</sup> time for central-radical groups, and in polynomial time for several prominent subclasses of central-radical groups. We also solve GPI in n <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">O(log log n)</sup> -time for groups whose solvable normal subgroups are elementary abelian but not necessarily central. As far as we are aware, this is the first time that a nontrivial algorithm with worst-case guarantees has tackled both aspects of GPI-actions and cohomology-simultaneously. Prior to this work, only n <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">O(log n)</sup> -time algorithms were known, even for groups with a central radical of constant size, such as Z(G) = Z_2. To develop these algorithms we utilize several mathematical results on the detailed structure of cohomology classes, as well as algorithmic results for code equivalence, coset intersection and cyclicity testing of modules over finite-dimensional associative algebras. We also suggest several promising directions for future work.

The isomorphism problem for finite groups of order $n$ (GpI) has long been known to be solvable in $n^{\log n+O(1)}$ time, but only recently were polynomial-time algorithms designed for several interesting group classes. Inspired by recent progress, we revisit the strategy for GpI via the extension theory of groups. The extension theory describes how a normal subgroup $N$ is related to $G/N$ via $G$, and this naturally leads to a divide-and-conquer strategy that “splits” GpI into two subproblems: one regarding group actions on other groups, and one regarding group cohomology. When the normal subgroup $N$ is abelian, this strategy is well known. Our first contribution is to extend this strategy to handle the case when $N$ is not necessarily abelian. This allows us to provide a unified explanation of all recent polynomial-time algorithms for special group classes. Guided by this strategy, to make further progress on GpI, we consider central-radical groups, proposed in Babai et al. [Code equivalence and group isomorphism, in Proceedings of the 22nd Annual ACM--SIAM Symposium on Discrete Algorithms (SODA'11), SIAM, Philadelphia, 2011, ACM, New York, pp. 1395--1408]: the class of groups such that $G$ modulo its center has no abelian normal subgroups. This class is a natural extension of the group class considered by Babai et al. [Polynomial-time isomorphism test for groups with no abelian normal subgroups (extended abstract), in International Colloquium on Automata, Languages, and Programming (ICALP), 2012, pp. 51--62], namely those groups with no abelian normal subgroups. Following the above strategy, we solve GpI in $n^{O(\log \log n)}$ time for central-radical groups, and in polynomial time for several prominent subclasses of central-radical groups. We also solve GpI in $n^{O(\log\log n)}$ time for groups whose solvable normal subgroups are elementary abelian but not necessarily central. As far as we are aware, this is the first time there have been worst-case guarantees on an $n^{o(\log n)}$-time algorithm that tackles both aspects of GpI---actions and cohomology---simultaneously. Prior to this work, the best proven upper bounds on algorithms for groups with central radicals were $n^{O(\log n)}$, even for groups with a central radical of constant size, such as ${Rad}(G) = Z(G)=\mathbb{Z}_2$. To develop our new algorithms we utilize several mathematical results on the detailed structure of cohomology classes, as well as algorithmic results for code equivalence, coset intersection, and cyclicity testing of modules over finite-dimensional associative algebras. We also suggest several promising directions for future work.

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

We prove new complexity results for computational problems in certain wreath products of groups and (as an application) for free solvable groups. For a finitely generated group we study the so-called power word problem (does a given expression u₁^{k₁} … u_d^{k_d}, where u₁, …, u_d are words over the group generators and k₁, …, k_d are binary encoded integers, evaluate to the group identity?) and knapsack problem (does a given equation u₁^{x₁} … u_d^{x_d} = v, where u₁, …, u_d,v are words over the group generators and x₁,…,x_d are variables, have a solution in the natural numbers). We prove that the power word problem for wreath products of the form G ≀ ℤ with G nilpotent and iterated wreath products of free abelian groups belongs to TC⁰. As an application of the latter, the power word problem for free solvable groups is in TC⁰. On the other hand we show that for wreath products G ≀ ℤ, where G is a so called uniformly strongly efficiently non-solvable group (which form a large subclass of non-solvable groups), the power word problem is coNP-hard. For the knapsack problem we show NP-completeness for iterated wreath products of free abelian groups and hence free solvable groups. Moreover, the knapsack problem for every wreath product G ≀ ℤ, where G is uniformly efficiently non-solvable, is Σ₂^p-hard.

Using the notion of isoclinism introduced by P. Hall for finite p-groups, we show that many important classes of finite p-groups have stable cohomology detected by abelian subgroups (see Theorem 11). Moreover, we show that the stable cohomology of the n-fold wreath product \(G_{n} = \mathbb{Z}/p \wr \ldots \wr \mathbb{Z}/p\) of cyclic groups \(\mathbb{Z}/p\) is detected by elementary abelian p-subgroups and we describe the resulting cohomology algebra explicitly. Some applications to the computation of unramified and stable cohomology of finite groups of Lie type are given.KeywordsStable CohomologyIterated Wreath ProductAbelian SubgroupFinite GroupAlgebra CohomologyThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Let [Formula: see text], [Formula: see text] be a sequence of groups each of which is either an alternating group, a symmetric group or a cyclic group. Let us construct a sequence [Formula: see text] of wreath products via [Formula: see text] and, for each [Formula: see text], [Formula: see text] via the natural permutation action. We determine the minimum number [Formula: see text] of generators required for each wreath product in this sequence.