Abstract
A group G is said to be of finite rank r if every finitely generated subgroup of G can be generated by at most r elements, and r is the least positive integer with a such property. If there is no such r, then the group G is said to be of infinite rank. In the present paper, it is proved that if G is an \(\mathfrak{X}\)-group of infinite rank whose proper subgroups of infinite rank are finite-by-hypercentral (respectively, hypercentral-by-finite), then all proper subgroups of G are finite-by-hypercentral (respectively, hypercentral-by-finite), where \(\mathfrak{X}\) is the class defined by N.S. Černikov as the closure of the class of periodic locally graded groups by the closure operations \(\boldsymbol{\acute{P}}\), \(\boldsymbol{\grave{P}}\) and \(\boldsymbol{L}\).
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This work was completed with the support of the General Directorate of Scientific Research and Technological Development (DGRSDT), Algeria.
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DILMI, A., TRABELSI, N. GROUPS WHOSE PROPER SUBGROUPS OF INFINITE RANK ARE FINITE-BY-HYPERCENTRAL OR HYPERCENTRAL-BY-FINITE. Acta Math. Hungar. 167, 492–500 (2022). https://doi.org/10.1007/s10474-022-01250-1
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DOI: https://doi.org/10.1007/s10474-022-01250-1