On Torsion-by-Nilpotent Groups
نویسنده
چکیده
Let C be a class of groups, closed under taking subgroups and quotients. We prove that if all metabelian groups of C are torsion-by-nilpotent, then all soluble groups of C are torsion-by-nilpotent. From that, we deduce the following consequence, similar to a well-known result of P. Hall: if H is a normal subgroup of a group G such that H and G/H ′ are (locally finite)-by-nilpotent, then G is (locally finite)-by-nilpotent. We give an example showing that this last statement is false when ”(locally finite)-by-nilpotent” is replaced by ”torsion-by-nilpotent”.
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