Groupes dont tous les sous-groupes propres vérifient une propriété donnée

Abstract

If X is a class of groups, then a group G is said to be minimal non-X if all its proper subgroups are in X, but G itself is not in X. Newman et Wiegold proved that an infinitely generated minimal non-nilpotent groups without maximal subgroups are metabelien Chernikov p-group, where p is a prime. In 2007, we are proved that there are no minimal non-(locally finite)-by-nilpotent groups which are infinitely generated. In this thesis we generalize these results by proving that an infinitely generated minimal non-(locally π-finite)-by-nilpotent groups are exactly a minimal non-nilpotent p-groups, where pπ is a prime and π is a set of primes. Also we prove that if G is a strongly locally graded F-perfect group of infinite rank whose proper subgroups of infinite rank are (locally π-finite)-by-(nilpotent of class at most c), then G itself is (locally π-finite)-by-(nilpotent of class at most c) if G is imperfect or without simple homomorphic image of infinite rank