Some group properties associated with twovariable words

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Soggetto
Group; Twovariable word; Engel conditionsAbstract
Let w(x; y) be a word in two variables and W the variety determined
by w. In this thesis, which includes a work made in collaboration with
C. Nicotera [5], we raise the following question: if for every pair of
elements a; b in a group G there exists g 2 G such that w(ag; b) = 1,
under what conditions does the group G belong to W ?
We introduce for every g 2 G the sets
Ww
L (g) = fa 2 G j w(g; a) = 1g
and
Ww
R (g) = fa 2 G j w(a; g) = 1g ;
where the letters L and R stand for left and right. In [2], M. Herzog, P.
Longobardi and M. Maj observed that if a group G belongs to the class
Y of all groups which cannot be covered by conjugates of any proper
subgroup, then G is abelian if for every a; b 2 G there exists g 2 G for
which [ag; b] = 1. Hence when G is a Y group and w is the commutator
word [x; y], the set Ww
L (g) = Ww
R (g) is the centralizer of g in G, and the
answer to the problem is a rmative. If G belongs to the class Y , we
show that, more generally, the problem has a positive answer whenever
each subset Ww
L (g) is a subgroup of G, or equivalently, if each subset
Ww
R (g) is a subgroup of G. The sets Ww
L (g) and Ww
R (g) can be called
the centralizerlike subsets associated with the word w. They need not
be subgroups in general: we examine some su cient conditions on the
group G ensuring that the sets Ww
L (g) and Ww
R (g) are subgroups of
G for all g in G. We denote by W w
L and W w
R respectively the class
of all groups G for which the set Ww
L (g) is a subgroup of G for every
g 2 G and the class of all groups G for which each subset Ww
R (g) is a subgroup... [edited by author]
Descrizione
2011  2012
Collections
Data
20130403Autore
Meriano, Maurizio
Metadata
Mostra tutti i dati dell'itemAutori  Meriano, Maurizio  
Data Realizzazione  20140210T10:57:11Z  
Date Disponibilità  20140210T10:57:11Z  
Data di Pubblicazione  20130403  
Identificatore (URI)  http://hdl.handle.net/10556/1009  
Identificatore (URI)  http://dx.doi.org/10.14273/unisa3  
Descrizione  2011  2012  en_US 
Abstract  Let w(x; y) be a word in two variables and W the variety determined by w. In this thesis, which includes a work made in collaboration with C. Nicotera [5], we raise the following question: if for every pair of elements a; b in a group G there exists g 2 G such that w(ag; b) = 1, under what conditions does the group G belong to W ? We introduce for every g 2 G the sets Ww L (g) = fa 2 G j w(g; a) = 1g and Ww R (g) = fa 2 G j w(a; g) = 1g ; where the letters L and R stand for left and right. In [2], M. Herzog, P. Longobardi and M. Maj observed that if a group G belongs to the class Y of all groups which cannot be covered by conjugates of any proper subgroup, then G is abelian if for every a; b 2 G there exists g 2 G for which [ag; b] = 1. Hence when G is a Y group and w is the commutator word [x; y], the set Ww L (g) = Ww R (g) is the centralizer of g in G, and the answer to the problem is a rmative. If G belongs to the class Y , we show that, more generally, the problem has a positive answer whenever each subset Ww L (g) is a subgroup of G, or equivalently, if each subset Ww R (g) is a subgroup of G. The sets Ww L (g) and Ww R (g) can be called the centralizerlike subsets associated with the word w. They need not be subgroups in general: we examine some su cient conditions on the group G ensuring that the sets Ww L (g) and Ww R (g) are subgroups of G for all g in G. We denote by W w L and W w R respectively the class of all groups G for which the set Ww L (g) is a subgroup of G for every g 2 G and the class of all groups G for which each subset Ww R (g) is a subgroup... [edited by author]  en_US 
Lingua  en  en_US 
Editore  Universita degli studi di Salerno  en_US 
Soggetto  Group  en_US 
Soggetto  Twovariable word  en_US 
Soggetto  Engel conditions  en_US 
Titolo  Some group properties associated with twovariable words  en_US 
Tipo  Doctoral Thesis  en_US 
MIUR  MAT/02 ALGEBRA  en_US 
Coordinatore  Longobardi, Patrizia  en_US 
Ciclo  XI n.s.  en_US 
Tutor  Longobardi, Patrizia  en_US 
CoTutor  Nicotera, Chiara  en_US 
Dipartimento  Matematica  en_US 