scholarly journals BEING CAYLEY AUTOMATIC IS CLOSED UNDER TAKING WREATH PRODUCT WITH VIRTUALLY CYCLIC GROUPS

2021 ◽  
pp. 1-11
Author(s):  
DMITRY BERDINSKY ◽  
MURRAY ELDER ◽  
JENNIFER TABACK
Keyword(s):  
Computer Science ◽  
Wreath Product ◽  
Cyclic Group ◽  
Wreath Products ◽  
Infinite Cyclic Group ◽  
Cyclic Groups ◽  
Automatic Groups

Abstract We extend work of Berdinsky and Khoussainov [‘Cayley automatic representations of wreath products’, International Journal of Foundations of Computer Science27(2) (2016), 147–159] to show that being Cayley automatic is closed under taking the restricted wreath product with a virtually infinite cyclic group. This adds to the list of known examples of Cayley automatic groups.

Automorphisms of Iterated Wreath Product p-Groups

2012 ◽  
Vol 55 (2) ◽  
pp. 390-399 ◽  
Author(s):  
Jeffrey M. Riedl
Keyword(s):  
Automorphism Group ◽  
Representation Theory ◽  
Wreath Product ◽  
Wreath Products ◽  
Cyclic Groups ◽  
Important Family

AbstractWe determine the order of the automorphism group Aut(W) for each member W of an important family of finite p-groups that may be constructed as iterated regular wreath products of cyclic groups. We use a method based on representation theory.

scholarly journals Generalized Pattern Avoidance Condition for the Wreath Product of Cyclic Groups with Symmetric Groups

2013 ◽  
Vol 2013 ◽  
pp. 1-17
Author(s):  
Sergey Kitaev ◽  
Jeffrey Remmel ◽  
Manda Riehl
Keyword(s):  
Symmetric Group ◽  
Wreath Product ◽  
Cyclic Group ◽  
Generating Functions ◽  
Symmetric Groups ◽  
Pattern Avoidance ◽  
Catalan Numbers ◽  
Cyclic Groups ◽  
Bijective Proof ◽  
Avoidance Condition

We continue the study of the generalized pattern avoidance condition for Ck≀Sn, the wreath product of the cyclic group Ck with the symmetric group Sn, initiated in the work by Kitaev et al., In press. Among our results, there are a number of (multivariable) generating functions both for consecutive and nonconsecutive patterns, as well as a bijective proof for a new sequence counted by the Catalan numbers.

Wreath Product of O*-Groups that is not in O*

1971 ◽  
Vol 14 (3) ◽  
pp. 453-454
Author(s):  
S. V. Modak
Keyword(s):  
Wreath Product ◽  
Cyclic Group ◽  
Metabelian Group ◽  
Ordered Group ◽  
Infinite Cyclic Group ◽  
Free Metabelian Group ◽  
Ordered Groups

It is well known that the wreath product of two ordered groups is an ordered group. In [2] Fuchs asks if the same is true for O*-groups. Here we construct an example to show that the wreath product of an infinite cyclic group with a free metabelian group is not an O*-group.

ON THE TOTAL COMPONENT OF THE PARTIAL SCHUR MULTIPLIER

2016 ◽  
Vol 100 (3) ◽  
pp. 374-402 ◽  
Author(s):  
H. G. G. DE LIMA ◽  
H. PINEDO
Keyword(s):  
Direct Product ◽  
Cyclic Group ◽  
Closed Field ◽  
Schur Multiplier ◽  
Dihedral Groups ◽  
Infinite Cyclic Group ◽  
Algebraically Closed Field ◽  
Cyclic Groups

In this paper we determine the structure of the total component of the Schur multiplier over an algebraically closed field of some relevant families of groups, such as dihedral groups, dicyclic groups, the infinite cyclic group and the direct product of two finite cyclic groups.

scholarly journals Cayley Automatic Representations of Wreath Products

2016 ◽  
Vol 27 (02) ◽  
pp. 147-159 ◽  
Author(s):  
Dmitry Berdinsky ◽  
Bakhadyr Khoussainov
Keyword(s):  
Wreath Product ◽  
Lower Bounds ◽  
Finite Automata ◽  
Cayley Graphs ◽  
Wreath Products ◽  
Upper And Lower Bounds ◽  
Automatic Groups ◽  
Pushdown Automata

We construct the representations of Cayley graphs of wreath products using finite automata, pushdown automata and nested stack automata. These representations are in accordance with the notion of Cayley automatic groups introduced by Kharlampovich, Khoussainov and Miasnikov and its extensions introduced by Elder and Taback. We obtain the upper and lower bounds for a length of an element of a wreath product in terms of the representations constructed.

ON GENERATION OF WREATH PRODUCTS OF CYCLIC GROUPS BY TWO STATE TIME VARYING MEALY AUTOMATA

2006 ◽  
Vol 16 (02) ◽  
pp. 397-415 ◽  
Author(s):  
ADAM WORYNA
Keyword(s):  
Topological Group ◽  
Wreath Product ◽  
Wreath Products ◽  
Time Varying ◽  
State Time ◽  
Cyclic Groups ◽  
Mealy Automaton ◽  
Mealy Automata

We proof that an infinitely iterated wreath product of finite cyclic groups of pairwise coprime orders is generated as a topological group by two elements a and b. The group G = 〈a, b〉 may be represented by a 2-state time-varying Mealy automaton. We derive some properties of G.

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.

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

scholarly journals Semigroups with few endomorphisms

1969 ◽  
Vol 10 (1-2) ◽  
pp. 162-168 ◽  
Author(s):  
Vlastimil Dlab ◽  
B. H. Neumann
Keyword(s):  
Cyclic Group ◽  
Finite Groups ◽  
Automorphism Groups ◽  
Infinite Groups ◽  
Infinite Cyclic Group

Large finite groups have large automorphism groups [4]; infinite groups may, like the infinite cyclic group, have finite automorphism groups, but their endomorphism semigroups are infinite (see Baer [1, p. 530] or [2, p. 68]). We show in this paper that the corresponding propositions for semigroups are false.

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