Groups with pronormal deviation
A subgroup \(X\) of a group \(G\) is said to be pronormal if \(X\) and \(X^{g}\) are conjugate in \(\langle X, X^{g} \rangle\) for every element \(g\in G\). Normal subgroups and maximal subgroups are obvious examples of pronormal subgroups in arbitrary groups; moreover, Sylow subgroups of finite groups and Hall subgroups of finite soluble groups are always pronormal. The main result proved in this article is the following Theorem: Let \(G\) be a radical group with pronormal deviation. Then either \(G\) is minimax or all its subgroups are pronormal. In particular, the pronormal deviation of \(G\) is at most 1. (An explanation of the technical terms can be found in the paper.)
scientific article; zbMATH DE number 7603201