We consider a finiteness condition on centralizers in a group G, namely that |C_G(x):⟨x⟩|<∞ for every ⟨x⟩⋪G. For periodic groups, this is the same as |C_G(x)|<∞ for every ⟨x⟩⋪G. We give a full description of locally finite groups satisfying this condition. As it turns out, they are a special type of cyclic extensions of Dedekind groups. We also study a variation of our condition, where the requirement of finiteness is replaced with a bound: |C_G(x):⟨x⟩|≤n for every ⟨x⟩⋪G, for some fixed n. In this case, we are able to extend our analysis to the class of periodic locally graded groups. The research that led to the present paper was partially supported by a grant of the group GNSAGA of INdAM.

A finitiness condition on centralizers in locally finite groups

TORTORA, ANTONIO;TOTA, Maria
2017

Abstract

We consider a finiteness condition on centralizers in a group G, namely that |C_G(x):⟨x⟩|<∞ for every ⟨x⟩⋪G. For periodic groups, this is the same as |C_G(x)|<∞ for every ⟨x⟩⋪G. We give a full description of locally finite groups satisfying this condition. As it turns out, they are a special type of cyclic extensions of Dedekind groups. We also study a variation of our condition, where the requirement of finiteness is replaced with a bound: |C_G(x):⟨x⟩|≤n for every ⟨x⟩⋪G, for some fixed n. In this case, we are able to extend our analysis to the class of periodic locally graded groups. The research that led to the present paper was partially supported by a grant of the group GNSAGA of INdAM.
File in questo prodotto:
Non ci sono file associati a questo prodotto.

I documenti in IRIS sono protetti da copyright e tutti i diritti sono riservati, salvo diversa indicazione.

Utilizza questo identificativo per citare o creare un link a questo documento: http://hdl.handle.net/11386/4676310
 Attenzione

Attenzione! I dati visualizzati non sono stati sottoposti a validazione da parte dell'ateneo

Citazioni
  • ???jsp.display-item.citation.pmc??? ND
  • Scopus 3
  • ???jsp.display-item.citation.isi??? 2
social impact