zbMATH — the first resource for mathematics

Groups with dense pronormal subgroups. (English) Zbl 0754.20011
Let \(\chi\) be a property pertaining to subgroups. A group \(G\) is said to have dense (respectively: weakly dense) \(\chi\)-subgroups if for each pair \((H,K)\) of subgroups of \(G\) such that \(H<K\) and \(H\) is not maximal in \(K\), there exists a \(\chi\)-subgroup \(X\) of \(G\) with \(H<X<K\) (respectively: \(H\leq X\leq K)\). In this paper groups with dense pronormal subgroups are considered. Here a subgroup \(P\) of a group \(G\) is called pronormal if for each element \(x\) of \(G\) the subgroups \(P\) and \(P^ x\) are conjugate in \(\langle P,P^ x\rangle\). Improving a result of A. Mann [Isr. J. Math. 6, 13-25 (1968; Zbl 0155.050)] the author shows that a locally soluble non-periodic group with weakly dense pronormal subgroups is abelian. Moreover, it is proved that a locally finite group with dense pronormal subgroups is hypercyclic and its commutator subgroup is nilpotent. This last result does not hold for groups with weakly dense pronormal subgroups, as the alternating group of degree 5 has this property.

20F16 Solvable groups, supersolvable groups
20F50 Periodic groups; locally finite groups
20E25 Local properties of groups
20E07 Subgroup theorems; subgroup growth