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

Authors: FRANCESCO DE GIOVANNI, FEDERICA SACCOMANNO

Abstract: A group G is said to have the CF-property if the index |X:X_G| 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.

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

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