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.

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.

scholarly journals On varieties of soluble groups II

1972 ◽  
Vol 7 (3) ◽  
pp. 437-441 ◽  
Author(s):  
J.R.J. Groves
Keyword(s):  
Nilpotent Group ◽  
Metabelian Groups ◽  
Locally Finite ◽  
Soluble Groups ◽  
Finite Exponent

It is shown that, in a variety which does not contain all metabelian groups and is contained in a product of (finitely many) varieties each of which is soluble or locally finite, every group is an extension of a group of finite exponent by a nilpotent group by a group of finite exponent.

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.

scholarly journals Minimal generating sets for some wreath products of groups

1973 ◽  
Vol 9 (1) ◽  
pp. 127-136
Author(s):  
Yeo Kok Chye
Keyword(s):  
Abelian Group ◽  
Nilpotent Group ◽  
Wreath Product ◽  
Wreath Products ◽  
Finitely Generated ◽  
Generating Set ◽  
Generating Sets ◽  
Products Of Groups ◽  
Minimal Generating Set ◽  
Standard Wreath Product

Let d(G) denote the minimum of the cardinalities of the generating sets of the group G. Call a generating set of cardinality d(G) a minimal generating set for G. If A is a finitely generated nilpotent group, B a non-trivial finitely generated abelian group and A wr B is their (restricted, standard) wreath product, then it is proved (by explicitly constructing a minimal generating set for A wr B ) that d(AwrB) = max{l+d(A), d(A×B)} where A × B is their direct product.

An example in the theory of solubl groups

1970 ◽  
Vol 67 (1) ◽  
pp. 13-16 ◽  
Author(s):  
T. O. Hawkes
Keyword(s):  
Soluble Group ◽  
Nilpotent Groups ◽  
Saturated Formation ◽  
Finite Soluble Group ◽  
Carter Subgroup ◽  
Soluble Groups ◽  
University Of Wisconsin ◽  
Unpublished Work ◽  
The University

Let G be a finite soluble group. In (1) Alperin proves that two system normalizers of G contained in the same Carter subgroup C of G are conjugate in C. In recent unpublished work G.A.Chambers of the University of Wisconsin has proved that, if is a saturated formation, the -normalizers of an A-group are pronormal subgruops; hence, in particular, that two -normalizers contained in an -projector E of an A-group are conjugate in E. In this note we describe an example which shows that in Alperin's theorem the class of nilpotent groups cannot in general be replaced by an arbitary saturated formation without some restriction on the class of soluble groups under consideration. we provePROPOSITION. There exists a saturated formationand a group G which has two-normalizers E1and E2contained in an-projector F of G such that E1and E2are not conjugate in F.

CLT groups and wreath products

1987 ◽  
Vol 42 (2) ◽  
pp. 183-195 ◽  
Author(s):  
Rolf Brandl
Keyword(s):  
Nilpotent Group ◽  
Wreath Product ◽  
Wreath Products ◽  
Splitting Field ◽  
Odd Order ◽  
Group Belonging

AbstractIn this paper the question is considered of when the wreath product of a nilpotent group with a CLT group G is a CLT group. It is shown that if the field with Pr elements is a splitting field of a Hall P1–subgroup of G, then P wr G is a CLT group for all p–groups P with |P/P1|≥ pr. Moreover, the class of all groups G having the property that N wr G is a CLT group for every nilpotent group N is shown to be quite large. For exmple, every group of odd order can be embedded as a subgroup of a group belonging to this class.

scholarly journals On central automorphisms of finite-by-nilpotent groups

1990 ◽  
Vol 33 (2) ◽  
pp. 191-201 ◽  
Author(s):  
Silvana Franciosi ◽  
Francesco de Giovanni
Keyword(s):  
Nilpotent Group ◽  
Finite Order ◽  
Nilpotent Groups ◽  
Finiteness Condition ◽  
Factor Group ◽  
Central Automorphism ◽  
Central Automorphisms ◽  
Finite Exponent

The effect of imposing a certain finiteness condition on the group of central automorphisms of a finite-by-nilpotent group is investigated. In particular it is shown that, if each central automorphism of a finite-by-nilpotent group G has finite order, then the factor group G/Z(G) has finite exponent.

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 .

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

2014 ◽  
Vol 13 (05) ◽  
pp. 1350156 ◽  
Author(s):  
ZHENGXING LI ◽  
JINKE HAI
Keyword(s):  
Finite Group ◽  
Nilpotent Group ◽  
Direct Consequence ◽  
Nilpotent Groups ◽  
Wreath Products ◽  
Coleman Automorphism ◽  
A Finite Group ◽  
Generalized Quaternion ◽  
Standard Wreath Product ◽  
The Normalizer Property

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.

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

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