Subgroups of finite index in T-groups. (English) Zbl 0598.20027

A group in which every subnormal subgroup is normal is called a T-group. By a result of Gaschütz every subgroup of a finite soluble T-group is again a T-group. On the other hand, it is not difficult to construct an infinite soluble T-group with subgroups which are not T-groups. However, a subgroup of finite index of an arbitrary soluble T-group will be a T- group; this is immediate from Theorem A: If G is a T-group, then every subgroup of finite index of G containing some term of the derived series of G, is again a T-group. For general (i.e. not necessarily soluble) T- groups, it is shown (Theorem B) that corresponding to each positive integer n there is a positive integer f(n) such that every subgroup H of index n of any T-group G has the property that each subnormal subgroup of H has defect \(\leq f(n)\) in H. An example is given showing that the natural generalization of Theorem B to groups G in which every subnormal subgroup has defect \(\leq k\) where \(k>1\), is not valid, and a further, more technical result, analogous to Theorem A, is proved. The paper is very clearly and interestingly written.
Reviewer: R.G.Burns


20E15 Chains and lattices of subgroups, subnormal subgroups
20F16 Solvable groups, supersolvable groups
20E07 Subgroup theorems; subgroup growth
20F14 Derived series, central series, and generalizations for groups
20F22 Other classes of groups defined by subgroup chains