Abstract
It is well-known that every virtually abelian group contains an abelian characteristic subgroup of finite index. We shall say that a group class \(\mathfrak {X}\) is F-characteristic if any group containing an \(\mathfrak {X}\)-subgroup of finite index has also a characteristic subgroup of finite index that belongs to \(\mathfrak {X}\). Thus the class \(\mathfrak {A}\) of abelian groups is F-characteristic. The aim of this paper is to prove that many interesting classes of infinite groups are F-characteristic. Moreover, it is shown that the class of free groups and that of free abelian groups are not F-characteristic.
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de Giovanni, F., Trombetti, M. Large Characteristic Subgroups with Restricted Conjugacy Classes. Results Math 74, 166 (2019). https://doi.org/10.1007/s00025-019-1089-5
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DOI: https://doi.org/10.1007/s00025-019-1089-5