Centralizers in Locally Finite Groups

Pavel SHUMYATSKY

Makale detayları

Centralizers in Locally Finite Groups

Yazarlar:

Pavel SHUMYATSKY

Yayıncı:

TUBITAK

Dergi:

Turkish Journal of Mathematics 2007 Vol.31 Issue Suppl, pp.149-170

Belge Tipi:

Makale

Yayın Tarihi:

11/2006

Belge Dili:

Türkçe

Document ID (Org No.):

2000748505

Sıklık:

İki Aylık

eISSN:

1303-6149

pISSN:

1300-0098

Anahtar kelimeler:

Özet:

 

The topic of the present paper is the following question.    Let G be a locally finite group admitting an automorphism f of  finite order such that the centralizer CG(f) satisfies certain  finiteness conditions.  What impact does this have on the structure of  the group G?    Equivalently, one can ask the same question when f is an element of  G. Sometimes the impact is quite strong and the paper is a survey of  results illustrating this phenomenon.  In particular, we concentrate  on results where G is shown to have a large nilpotent or soluble  subgroup.  Naturally, in each case the result depends on the order of  the automorphism f and kind of conditions imposed on CG(f).  We  shall be considering mostly the classical finiteness conditions such  as CG(f) being finite, Chernikov, and of finite rank,  respectively.  It is not a purpose of the paper to survey numerous  results on automorphisms of finite groups.  In particular, among  important topics that are left out of the present discussion are  p-automorphisms of p-groups\" (see [36]) and length problems\" (see  [73]).  However, in some situations (like, for example, when CG(f)  is finite) problems on infinite groups quickly reduce to finite groups  and in those cases working with finite groups is very natural.    A separate section of the paper is devoted to the case when f is of  order two, a prime, four, or other, respectively.  However, before  anything else we address in Section 1 the following related question.    Given a periodic group G with an automorphism f, what additional  assumptions on G and CG(f) ensure that G is locally finite?.

 

The topic of the present paper is the following question.    Let G be a locally finite group admitting an automorphism f of  finite order such that the centralizer CG(f) satisfies certain  finiteness conditions.  What impact does this have on the structure of  the group G?    Equivalently, one can ask the same question when f is an element of  G. Sometimes the impact is quite strong and the paper is a survey of  results illustrating this phenomenon.  In particular, we concentrate  on results where G is shown to have a large nilpotent or soluble  subgroup.  Naturally, in each case the result depends on the order of  the automorphism f and kind of conditions imposed on CG(f).  We  shall be considering mostly the classical finiteness conditions such  as CG(f) being finite, Chernikov, and of finite rank,  respectively.  It is not a purpose of the paper to survey numerous  results on automorphisms of finite groups.  In particular, among  important topics that are left out of the present discussion are  p-automorphisms of p-groups\" (see [36]) and length problems\" (see  [73]).  However, in some situations (like, for example, when CG(f)  is finite) problems on infinite groups quickly reduce to finite groups  and in those cases working with finite groups is very natural.    A separate section of the paper is devoted to the case when f is of  order two, a prime, four, or other, respectively.  However, before  anything else we address in Section 1 the following related question.    Given a periodic group G with an automorphism f, what additional  assumptions on G and CG(f) ensure that G is locally finite?.

Makale detayları

Centralizers in Locally Finite Groups

Yazarlar:

Pavel SHUMYATSKY

Yayıncı:

TUBITAK

Dergi:

Turkish Journal of Mathematics 2007 Vol.31 Issue Suppl, pp.149-170

Belge Tipi:

Makale

Yayın Tarihi:

11/2006

Belge Dili:

Türkçe

Document ID (Org No.):

2000748505

Sıklık:

İki Aylık

eISSN:

1303-6149

pISSN:

1300-0098

Locally finite groups

centralizer

Sayı

Makale detayları

Centralizers in Locally Finite Groups

Yazarlar: Pavel SHUMYATSKY

Yayıncı: TUBITAK

Dergi: Turkish Journal of Mathematics 2007 Vol.31 Issue Suppl, pp.149-170

Belge Tipi: Makale

Yayın Tarihi: 11/2006

Belge Dili: Türkçe

Document ID (Org No.): 2000748505

Sıklık: İki Aylık

eISSN: 1303-6149

pISSN: 1300-0098

Locally finite groups

centralizer

Sayı

Yazarlar:

Pavel SHUMYATSKY

Yayıncı:

TUBITAK

Dergi:

Turkish Journal of Mathematics 2007 Vol.31 Issue Suppl, pp.149-170

Belge Tipi:

Makale

Yayın Tarihi:

11/2006

Belge Dili:

Türkçe

Document ID (Org No.):

2000748505

Sıklık:

İki Aylık

eISSN:

1303-6149

pISSN:

1300-0098