Groups in which proper subgroups contain a nilpotent subgroup of finite index. (Italian. English summary) Zbl 0578.20027
The class of groups studied in this paper are the locally graded groups all of whose subgroups contain some nilpotent by finite group. The main result is Theorem 1. Let G be a periodic, locally graded and non locally nilpotent group. Then G is minimal non nilpotent by finite if and only if \(G=V\rtimes H\) where \(H=C_{p^{\infty}}\) for some prime p and for some prime \(q\neq p\) the group V is a special q-group, V’ is centralized by H and V/V’ is a minimal normal subgroup of G/V’.
The proof involves a series of reductions and case distinctions and reference to earlier work of the author. Furthermore a series of examples for the groups satisfying the hypothesis of Theorem 1 is constructed.
The proof involves a series of reductions and case distinctions and reference to earlier work of the author. Furthermore a series of examples for the groups satisfying the hypothesis of Theorem 1 is constructed.
Reviewer: H.R.Schneebeli
MSC:
| 20F19 | Generalizations of solvable and nilpotent groups |
| 20E25 | Local properties of groups |
| 20F50 | Periodic groups; locally finite groups |
| 20F22 | Other classes of groups defined by subgroup chains |
| 20E07 | Subgroup theorems; subgroup growth |
| 20E15 | Chains and lattices of subgroups, subnormal subgroups |
