Some group properties associated with twovariable words

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Subject
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]
xmlui.metadata.dc.description
2011  2012
Collections
Date
20130403Author
Meriano, Maurizio
Metadata
Show full item recordxmlui.metadata.dc.contributor.author  Meriano, Maurizio  
xmlui.metadata.dc.date.accessioned  20140210T10:57:11Z  
xmlui.metadata.dc.date.available  20140210T10:57:11Z  
xmlui.metadata.dc.date.issued  20130403  
xmlui.metadata.dc.identifier.uri  http://hdl.handle.net/10556/1009  
xmlui.metadata.dc.identifier.uri  http://dx.doi.org/10.14273/unisa3  
xmlui.metadata.dc.description  2011  2012  en_US 
xmlui.metadata.dc.description.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 
xmlui.metadata.dc.language.iso  en  en_US 
xmlui.metadata.dc.publisher  Universita degli studi di Salerno  en_US 
xmlui.metadata.dc.subject  Group  en_US 
xmlui.metadata.dc.subject  Twovariable word  en_US 
xmlui.metadata.dc.subject  Engel conditions  en_US 
xmlui.metadata.dc.title  Some group properties associated with twovariable words  en_US 
xmlui.metadata.dc.type  Doctoral Thesis  en_US 
xmlui.metadata.dc.subject.miur  MAT/02 ALGEBRA  en_US 
xmlui.metadata.dc.contributor.coordinatore  Longobardi, Patrizia  en_US 
xmlui.metadata.dc.description.ciclo  XI n.s.  en_US 
xmlui.metadata.dc.contributor.tutor  Longobardi, Patrizia  en_US 
xmlui.metadata.dc.contributor.cotutor  Nicotera, Chiara  en_US 
xmlui.metadata.dc.identifier.Dipartimento  Matematica  en_US 