Standard Wreath Product Research Articles

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.

Central automorphisms of standard wreath products

Central automorphisms of standard wreath products

Some closure results for 𝒞-approximable groups

We investigate closure results for C-approximable groups, for certain classes C of groups with invariant length functions. In particular we prove, each time for certain (but not necessarily the same) classes C that (i) the direct product of two C-approximable groups is C-approximable; (ii) the restricted standard wreath product G wr H is C-approximable when G is C-approximable and H is residually finite; and (iii) a group G$ with normal subgroup N is C-approximable when N is C-approximable and G/N is amenable. Our direct product result is valid for LEF, weakly sofic and hyperlinear groups, as well as for all groups that are approximable by finite groups equipped with commutator-contractive invariant length functions (considered in \cite{Thom}). Our wreath product result is valid for weakly sofic groups, and we prove it separately for sofic groups. We note that this last result has recently been generalised by Hayes and Sale, who prove that the restricted standard wreath product of any two sofic groups is sofic. Our result on extensions by amenable groups is valid for weakly sofic groups, and was proved by Elek and Szabo for sofic groups N.

COLEMAN AUTOMORPHISMS OF STANDARD WREATH PRODUCTS OF NILPOTENT GROUPS BY GROUPS WITH PRESCRIBED SYLOW 2-SUBGROUPS

Let G = N wr H be the standard wreath product of N by H, where N is a finite nilpotent group and H is a finite group whose Sylow 2-subgroups are either cyclic, dihedral or generalized quaternion. It is shown that every Coleman automorphism of G is inner. As a direct consequence of this result, it is obtained that the normalizer property holds for G.

Coleman automorphisms of standard wreath products of finite abelian groups by 2-closed groups

Coleman automorphisms of standard wreath products of finite abelian groups by 2-closed groups

A generalization of the first Malcev theorem on nilpotent semigroups and nilpotency of the wreath product of semigroups

We describe all [0-]simple semigroups that are nilpotent in the sense of Malcev. This generalizes the first Malcev theorem on nilpotent (in the sense of Malcev) semigroups. It is proved that if the extended standard wreath product of semigroups is nilpotent in the sense of Malcev and the passive semigroup is not nilpotent, then the active semigroup of the wreath product is a finite nilpotent group. In addition to that, the passive semigroup is uniform periodic. Necessary and sufficient conditions are found under which the extended standard wreath product of semigroups is nilpotent in the sense of Malcev in the case where each of the semigroups of the wreath product generates a variety of finite step.

The Normalizer Property for Integral Group Rings of Some Finite Nilpotent-by-Nilpotent Groups

In this article, it is shown that the normalizer property holds for the following two kinds of finite nilpotent-by-nilpotent groups: (1) G = NwrH is the standard wreath product of N by H, where N is a finite nilpotent group and H is a finite abelian 2-group; (2) G is a finite group having a normal nilpotent subgroup N such that the integral group ring ℤ(G/N) has only trivial units. Our results generalize a result of Yuanlin Li and extend some ones obtained by Juriaans, Miranda, and Robério.

Proper actions of wreath products and generalizations

We study stability properties of the Haagerup property and of coarse embeddability in a Hilbert space, under certain semidirect products. In particular, we prove that they are stable under taking standard wreath products. Our construction also allows for a characterization of subsets with relative Property T in a standard wreath product.

Complete finite semidirect products and wreath products

A complete group is one with a trivial center and with all automorphisms inner. This paper uses group cohomology to give a sufficient condition for a finite semidirect product \({G = N \rtimes H}\) with CG(N) ≤ N to be complete and proves a partial converse. These results are enough to fully characterize complete finite permutational wreath products and to specialize that characterization in the case of finite standard wreath products.

On Property (FA) for wreath products

AbstractWe characterize permutational wreath products with Property (FA). For instance, we show that the standard wreath product

Separability and Efficiency Under Standard Wreath Product in Terms of Cayley Graphs

Separability and Efficiency Under Standard Wreath Product in Terms of Cayley Graphs

Finitely Presented Wreath Products and Double Coset Decompositions

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 characterization above, we discuss the following question: which finitely generated groups can have a finitely generated subgroup with finitely many double cosets? The discussion involves properties related to the structure of maximal subgroups, and to the profinite topology.

GROUPS WITH CONTEXT-FREE CO-WORD PROBLEM

The class of co-context-free groups is studied. A co-context-free group is defined as one whose co-word problem (the complement of its word problem) is context-free. This class is larger than the subclass of context-free groups, being closed under the taking of finite direct products, restricted standard wreath products with context-free top groups, and passing to finitely generated subgroups and finite index overgroups. No other examples of co-context-free groups are known. It is proved that the only examples amongst polycyclic groups or the Baumslag–Solitar groups are virtually abelian. This is done by proving that languages with certain purely arithmetical properties cannot be context-free; this result may be of independent interest.

Probabilistic Generation of Wreath Products of Non-abelian Finite Simple Groups#

We show that the probability of generating an iterated standard wreath product of non-abelian finite simple groups is positive and tends to 1 as the order of the first simple group tends to infinity. This has the consequence that the profinite group which is the inverse limit of these iterated wreath products is positively finitely generated. Information depending on the Classification of Finite Simple Groups is used throughout.

Wreath products of cyclic p-groups as automorphism groups

We prove that if p is a prime and W is the standard wreath product of two nontrivial cyclic p-groups X and Y then W is isomorphic to the full automorphism group of some group if and only if | X | = 2 and | Y | is 2 or 4.

ON GROUPS WHICH ARE SYNTACTIC MONOIDS OF DETERMINISTIC CONTEXT-FREE LANGUAGES

The author shows that the class of groups which are syntactic monoids of deterministic context-free languages is closed under restricted standard wreath products with virtually free top groups. This provides an answer to a question of T. Herbst.

The efficiency of standard wreath product

AbstractLet ξ be the set of all finite groups that have efficient presentations. In this paper we give sufficient conditions for the standard wreath product of two ξ-groups to be a ξ-group.

Finitarily linear wreath products

AbstractWe 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.

On the isomorphism of wreath products of groups

The well-known Neumann theorem on the isomorphism of standard wreath products is generalized to the wreath products of an arbitrary transitive permutation group and an abstract group.

$n$-Complete standard wreath products

$n$-Complete standard wreath products