Francesco GIOVANNI DE, Federica SACCOMANNO

A note on infinite groups whose subgroups are close to be normal-by-finite

A group G is said to have the CF-property if the index |X:XG| is finite for every subgroup X of G. Extending previous results by Buckley, Lennox, Neumann, Smith, and Wiegold, it is proven here that if G is a locally graded group whose proper subgroups have the CF-property, then G is abelian-by-finite, provided that all its periodic sections are locally finite. Groups in which all proper subgroups of infinite rank have the CF-property are also studied.

Anahtar Kelimeler:

Normal-by-finite subgroup, CF-group, group of infinite rank

A note on infinite groups whose subgroups are close to be normal-by-finite

Keywords:

Normal-by-finite subgroup, CF-group, group of infinite rank,

PDF

___

Bruno B. On groups with abelian-by-finite proper subgroups. Boll Un Mat Ital 1984; 3: 797–807.
Bruno B, Phillips RE. A note on groups with nilpotent-by-finite proper subgroups. Arch Math (Basel) 1995; 65: 369–374.
Buckley J, Lennox JC, Neumann BH, Smith H, Wiegold J. Groups with all subgroups normal-by-finite. J Austral Math Soc Ser A 1995; 59: 384–398.
De Falco M, de Giovanni F, Musella C. Groups whose proper subgroups of infinite rank have a transitive normality relation. Mediterranean J Math 2013; 10: 1999–2006.
De Falco M, de Giovanni F, Musella C, Trabelsi N. Groups with restrictions on subgroups of infinite rank. Rev Mat Iberoamericana 2014; 30: 535–548.
De Falco M, de Giovanni F, Musella C, Trabelsi N. Groups whose proper subgroups of infinite rank have finite conjugacy classes. Bull Austral Math Soc 2014; 89: 41–48. de Giovanni F, Martusciello M, Rainone C. Locally finite groups whose subgroups have finite normal oscillation.
Bull Austral Math Soc 2014; 89: 479–487. de Giovanni F, Saccomanno F. A note on groups of infinite rank whose proper subgroups are abelian-by-finite.
Colloquium Math (in press). Dixon MR, Evans MJ, Smith H. Groups with all proper subgroups nilpotent-by-finite rank. Arch Math (Basel) 2000; 75: 81–91.
Robinson DJS. Finiteness Conditions and Generalized Soluble Groups. Berlin, Germany: Springer, 1972.
Smith H, Wiegold J. Locally graded groups with all subgroups normal-by-finite. J Austral Math Soc Ser A 1996; 60: 222-227.