scholarly journals Kp-series and varieties generated by wreath products of p-groups

2018 ◽  
Vol 28 (08) ◽  
pp. 1693-1703
Author(s):  
V. H. Mikaelian
Keyword(s):  
Prime Number ◽  
Wreath Product ◽  
Direct Product ◽  
Finite Groups ◽  
Abelian Groups ◽  
Wreath Products ◽  
Product Formula ◽  
Number Formula ◽  
Finite Exponent

Let [Formula: see text] be a nilpotent [Formula: see text]-group of finite exponent and [Formula: see text] be an abelian [Formula: see text]-group of finite exponent for a given prime number [Formula: see text]. Then the wreath product [Formula: see text] generates the variety [Formula: see text] if and only if the group [Formula: see text] contains a subgroup isomorphic to the direct product [Formula: see text] of countably many copies of the cycle [Formula: see text] of order [Formula: see text]. The obtained theorem continues our previous study of cases when [Formula: see text] holds for some other classes of groups [Formula: see text] and [Formula: see text] (abelian groups, finite groups, etc.).

Central series in wreath products

1967 ◽  
Vol 63 (3) ◽  
pp. 551-567 ◽  
Author(s):  
J. D. P. Meldrum
Keyword(s):  
Wreath Product ◽  
Abelian Groups ◽  
Wreath Products ◽  
Nilpotency Class ◽  
Central Series ◽  
Base Group ◽  
Finite Exponent

In this paper we study the structure of the α-central series of the nilpotent wreath product of two Abelian groups, the α-central series being the intersection of the upper central series with the base group. Let C = A wr B, the standard restricted wreath product of A and B. Then Baumslag(1) showed that C is nilpotent if and only if A is a nilpotent p-group of finite exponent and B is a finite p-group for the same prime p. In (5) Liebeck calculated the nilpotency class of the nilpotent wreath product of two Abelian groups. We obtain an expression for an element of the base group to belong to a given term of the upper central series.

Bounds for the class of nilpotent wreath products

1966 ◽  
Vol 62 (2) ◽  
pp. 165-169 ◽  
Author(s):  
Teresa Scruton
Keyword(s):  
Wreath Product ◽  
Direct Product ◽  
Wreath Products ◽  
Nilpotency Class ◽  
Cyclic Groups ◽  
Image Position ◽  
Finite Exponent

Introduction. In his paper ((1)), Baumslag has shown that the wreath product A wr B of a group A by a group B is nilpotent if and only if A is a nilpotent p-group of finite exponent and B is a finite p-group, the prime p being the same for both groups. Liebeck ((3)) has obtained the exact nilpotency class of A wr B when A and B are Abelian. Let A be an Abelian p -group of exponent pn and let B be a direct product of cyclic groups, whose orders are pβ1, …, pβn, with β1 ≤ β2 ≤ … βn. Then A wr B has nilpotency class .

On nilpotent wreath products

1970 ◽  
Vol 68 (1) ◽  
pp. 1-15 ◽  
Author(s):  
J. D. P. Meldrum
Keyword(s):  
Wreath Product ◽  
Direct Product ◽  
Wreath Products ◽  
Nilpotency Class ◽  
Cyclic Groups ◽  
Image Position ◽  
Finite Exponent

1. Introduction. The wreath product A wr B of a group A by a group B is nilpotent if and only if A is a nilpotent p-group of finite exponent and B is a finite p-group for the same prime p (Baumslag (1)). When A is an Abelian p-group of exponent pk, and B is the direct product of cyclic groups of orders pβ1, …, pβn and β1 ≥ β2 ≥ …, ≥ βn, then Liebeck has shown that the nilpotency class c of A wr B is

Pointwise bound for ℓ-torsion in class groups: Elementary abelian extensions

2020 ◽  
Vol 0 (0) ◽  
Author(s):  
Jiuya Wang
Keyword(s):  
Galois Group ◽  
Prime Number ◽  
Finite Groups ◽  
Upper Bound ◽  
Abelian Groups ◽  
Class Groups ◽  
Abelian Extensions ◽  
Pointwise Bound ◽  
First Case

AbstractElementary abelian groups are finite groups in the form of {A=(\mathbb{Z}/p\mathbb{Z})^{r}} for a prime number p. For every integer {\ell>1} and {r>1}, we prove a non-trivial upper bound on the {\ell}-torsion in class groups of every A-extension. Our results are pointwise and unconditional. This establishes the first case where for some Galois group G, the {\ell}-torsion in class groups are bounded non-trivially for every G-extension and every integer {\ell>1}. When r is large enough, the unconditional pointwise bound we obtain also breaks the previously best known bound shown by Ellenberg and Venkatesh under GRH.

Algebraic Structures Associated to Orbifold Wreath Products

2010 ◽  
Vol 8 (2) ◽  
pp. 323-338
Author(s):  
Carla Farsi ◽  
Christopher Seaton
Keyword(s):  
Hopf Algebra ◽  
Wreath Product ◽  
Finite Groups ◽  
Lie Group ◽  
Wreath Products ◽  
Closed Manifold ◽  
Algebraic Structures ◽  
Present Structure ◽  
Smooth Closed Manifold ◽  
Connected Lie Group

AbstractWe present structure theorems in terms of inertial decompositions for the wreath product ring of an orbifold presented as the quotient of a smooth, closed manifold by a compact, connected Lie group acting almost freely. In particular we show that this ring admits λ-ring and Hopf algebra structures both abstractly and directly. This generalizes results known for global quotient orbifolds by finite groups.

scholarly journals ON THE REGULARITY CONJECTURE FOR THE COHOMOLOGY OF FINITE GROUPS

2008 ◽  
Vol 51 (2) ◽  
pp. 273-284 ◽  
Author(s):  
David J. Benson
Keyword(s):  
Finite Group ◽  
Wreath Product ◽  
Finite Groups ◽  
Cohomology Ring ◽  
Symmetric Groups ◽  
Wreath Products ◽  
Characteristic P ◽  
Residue Classes ◽  
The Difference ◽  
A Finite Group

AbstractLet $K$ be a field of characteristic $p$ and let $G$ be a finite group of order divisible by $p$. The regularity conjecture states that the Castelnuovo–Mumford regularity of the cohomology ring $H^*(G,K)$ is always equal to 0. We prove that if the regularity conjecture holds for a finite group $H$, then it holds for the wreath product $H\wr\mathbb{Z}/p$. As a corollary, we prove the regularity conjecture for the symmetric groups $\varSigma_n$. The significance of this is that it is the first set of examples for which the regularity conjecture has been checked, where the difference between the Krull dimension and the depth of the cohomology ring is large. If this difference is at most 2, the regularity conjecture is already known to hold by previous work.For more general wreath products, we have not managed to prove the regularity conjecture. Instead we prove a weaker statement: namely, that the dimensions of the cohomology groups are polynomial on residue classes (PORC) in the sense of Higman.

Wreath products and p–groups

1959 ◽  
Vol 55 (3) ◽  
pp. 224-231 ◽  
Author(s):  
Gilbert Baumslag
Keyword(s):  
Nilpotent Group ◽  
Wreath Product ◽  
Soluble Group ◽  
Nilpotent Groups ◽  
Wreath Products ◽  
Soluble Groups ◽  
Finite Exponent

The wreath product is a useful method for constructing new soluble groups from given ones (cf. P. Hall (3)). Now although the wreath product of one soluble group by another is (obviously) always soluble, the corresponding result is no longer true for nilpotent groups. It is the object of § 3 of this note to determine precisely when the wreath product W of a non-trivial nilpotent group A by a non-trivial nilpotent group B is nilpotent; in fact I prove that W is nilpotent if and only if both A and B are (nilpotent) p–groups with A of finite exponent and B finite.

CHARACTERIZATIONS OF SOME WREATH PRODUCTS OF GROUPS BY THEIR ENDOMORPHISM SEMIGROUPS

2008 ◽  
Vol 18 (02) ◽  
pp. 243-255 ◽  
Author(s):  
PEETER PUUSEMP
Keyword(s):  
Abelian Group ◽  
Wreath Product ◽  
Cyclic Group ◽  
Abelian Groups ◽  
Finite Abelian Group ◽  
Wreath Products ◽  
Endomorphism Semigroup ◽  
Finite Abelian Groups ◽  
Products Of Groups

Let A be a cyclic group of order pn, where p is a prime, and B be a finite abelian group or a finite p-group which is determined by its endomorphism semigroup in the class of all groups. It is proved that under these assumptions the wreath product A Wr B is determined by its endomorphism semigroup in the class of all groups. It is deduced from this result that if A, B, A0,…, An are finite abelian groups and A0,…, An are p-groups, p prime, then the wreath products A Wr B and An Wr (…( Wr (A1 Wr A0))…) are determined by their endomorphism semigroups in the class of all groups.

THE CONJUGACY CLASSES OF A SUBGROUP $S_{n}^{m} : C_{m}$ OF Smn, prime m

2008 ◽  
Vol 18 (04) ◽  
pp. 705-717
Author(s):  
KENNETH ZIMBA ◽  
MERIAM RABOSHAKGA
Keyword(s):  
Symmetric Group ◽  
Wreath Product ◽  
Direct Product ◽  
Cyclic Group ◽  
Finite Groups ◽  
Conjugacy Classes ◽  
Character Table ◽  
Split Extension ◽  
Alternative Method ◽  
Positive Integers

The conjugacy classes of any group are important since they reflect some aspects of the structure of the group. The construction of the conjugacy classes of finite groups has been a subject of research for several authors. Let n,m be positive integers and [Formula: see text] be the direct product of m copies of the symmetric group Sn of degree n. Then [Formula: see text] is a subgroup of the symmetric group Smn of degree m × n. Let g∈Smn, of type [mn] where each m-cycle contains one symbol from each set of symbols in that order on which the copies of Sn act. Then g permutes the elements of the copies of Sn in [Formula: see text] and generates a cyclic group Cm = 〈g〉 of order m. The wreath product of Sn with Cm is a split extension or semi-direct product of [Formula: see text] by Cm, denoted by [Formula: see text]. It is clear that [Formula: see text] is a subgroup of the symmetric group Smn. In this paper we give a method similar to coset analysis for constructing the conjugacy classes of [Formula: see text], where m is prime. Apart from the fact that this is an alternative method for constructing the conjugacy classes of the group [Formula: see text], this method is useful in the construction of Fischer–Clifford matrices of the group [Formula: see text]. These Fischer–Clifford matrices are useful in the construction of the character table of [Formula: see text].

Concerning nilpotent wreath products

1962 ◽  
Vol 58 (3) ◽  
pp. 443-451 ◽  
Author(s):  
Hans Liebeck
Keyword(s):  
Lower Bound ◽  
Wreath Product ◽  
Nilpotent Groups ◽  
Wreath Products ◽  
Nilpotency Class ◽  
Simple Construction ◽  
Metabelian Groups ◽  
Engel Condition ◽  
Finite Exponent

In a recent paper, (2), Baumslag showed that a wreath product of A by B is nilpotent if and only if A and B are nilpotent p-groups for the same prime p, with A of finite exponent and B finite. We shall calculate the (nilpotency) class of such groups when A and B are Abelian. This provides a lower bound for the class in the general case. We give a simple construction for a set of non-nilpotent metabelian groups which satisfy a finite Engel condition. With Engel class defined by Definition 6·1, we show that there are nilpotent groups of arbitrarily large nilpotency class for which the nilpotency class is equal to the Engel class.

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