On the solvability of 2-groups. (Sur la résolubilité des 2-groupes.) (French) Zbl 0794.20050
Summary: Let \(d\) be an integer and let \(G\) be a 2-group of finite exponent whose \(d\)-generator subgroups are nilpotent of class at most a fixed integer \(c\). Free groups of infinite rank in the variety defined by \(X^ 4=1\) show that if \(d>2\) and \(c=3d-2\), \(G\) is not necessarily solvable. On the other hand, we prove that \(G\) is solvable if \(d>0\) and \(c=3d-3\).
MSC:
20F50 | Periodic groups; locally finite groups |
20F19 | Generalizations of solvable and nilpotent groups |
20E25 | Local properties of groups |
20F16 | Solvable groups, supersolvable groups |
20E10 | Quasivarieties and varieties of groups |