Some topics in the theory of generalized fcgroups

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Soggetto
FC gruppi; Teoria dei gruppiAbstract
A finiteness condition is a grouptheoretical property which is possessed
by all finite groups: thus it is a generalization of finiteness. This embraces an immensely wide collection of properties like, for example, finiteness, finitely
generated, the maximal condition and so on. There are also numerous finiteness
conditions which restrict, in some way, a set of conjugates or a set of
commutators in a group. Sometimes these restrictions are strong enough to
impose a recognizable structure on the group. R. Baer and B.H. Neumann were the first authors to discuss groups in which there is a limitation on the number of conjugates which an element may have. An element x of a group G is called FCelement of G if
x has only a finite number of conjugates in G, that is to say, if G : CG(x)
is finite or, equivalently, if the factor group G/CG(⟨x⟩G) is finite. It is a
basic fact that the FCelements always form a characteristic subgroup. An
FCelement may be thought as a generalization of an element of the center
of the group, because the elements of the latter type have just one conjugate.
For this reason the subgroup of all FCelements is called the FCcenter and,
clearly, always contains the center. A group G is called an FCgroup if it
equals its FCcenter, in other words, every conjugacy class of G is finite.
Prominent among the FCgroups are groups with center of finite index: in
such a group each centralizer must be of finite index, because it contains the
center. Of course in particular all abelian groups and all finite groups are
FCgroups. Further examples of FCgroups can be obtained by noting that
the class of FCgroups is closed with respect to forming subgroups, images
and direct products. The theory of FCgroups had a strong development in
the second half of the last century and relevant contributions have been given
by several important authors including R. Baer, B.H. Neumann, Y.M. Gorcakov, Chernikov,L.A. Kurdachenko, and
many others. We shall use the monographs , as a general reference for results on FCgroups. The study of FCgroups can be considered as a natural investigation on the properties common to both finite
groups and abelian groups.
A particular interest has been devoted to groups having many FCsubgroups
or many FCelements. [edited by the author]
Descrizione
2009  2010
Collections
Data
20110215Autore
Romano, Emanuela
Metadata
Mostra tutti i dati dell'itemAutori  Romano, Emanuela  
Data Realizzazione  20111109T16:06:04Z  
Date Disponibilità  20111109T16:06:04Z  
Data di Pubblicazione  20110215  
Identificatore (URI)  http://hdl.handle.net/10556/174  
Descrizione  2009  2010  en_US 
Abstract  A finiteness condition is a grouptheoretical property which is possessed by all finite groups: thus it is a generalization of finiteness. This embraces an immensely wide collection of properties like, for example, finiteness, finitely generated, the maximal condition and so on. There are also numerous finiteness conditions which restrict, in some way, a set of conjugates or a set of commutators in a group. Sometimes these restrictions are strong enough to impose a recognizable structure on the group. R. Baer and B.H. Neumann were the first authors to discuss groups in which there is a limitation on the number of conjugates which an element may have. An element x of a group G is called FCelement of G if x has only a finite number of conjugates in G, that is to say, if G : CG(x) is finite or, equivalently, if the factor group G/CG(⟨x⟩G) is finite. It is a basic fact that the FCelements always form a characteristic subgroup. An FCelement may be thought as a generalization of an element of the center of the group, because the elements of the latter type have just one conjugate. For this reason the subgroup of all FCelements is called the FCcenter and, clearly, always contains the center. A group G is called an FCgroup if it equals its FCcenter, in other words, every conjugacy class of G is finite. Prominent among the FCgroups are groups with center of finite index: in such a group each centralizer must be of finite index, because it contains the center. Of course in particular all abelian groups and all finite groups are FCgroups. Further examples of FCgroups can be obtained by noting that the class of FCgroups is closed with respect to forming subgroups, images and direct products. The theory of FCgroups had a strong development in the second half of the last century and relevant contributions have been given by several important authors including R. Baer, B.H. Neumann, Y.M. Gorcakov, Chernikov,L.A. Kurdachenko, and many others. We shall use the monographs , as a general reference for results on FCgroups. The study of FCgroups can be considered as a natural investigation on the properties common to both finite groups and abelian groups. A particular interest has been devoted to groups having many FCsubgroups or many FCelements. [edited by the author]  en_US 
Lingua  en  en_US 
Editore  Universita degli studi di Salerno  en_US 
Soggetto  FC gruppi  en_US 
Soggetto  Teoria dei gruppi  en_US 
Titolo  Some topics in the theory of generalized fcgroups  en_US 
Tipo  Doctoral Thesis  en_US 
MIUR  MAT/02 ALGEBRA  en_US 
Coordinatore  Longobardi, Patrizia  en_US 
Ciclo  IX n.s.  en_US 
Tutor  Vincenzi, Giovanni  en_US 
Dipartimento  Matematica e Informatica  en_US 