Dual-standard subgroups of finite and locally finite groups

A subgroup D of a group G is called dual-standard if, for all subgroups X and Y of G, =χD. When G is finite, Zappa has given some information concerning the way in which D is embedded in G and the structure of G itself. Among other things, Zappa makes reference to the maximal normal Hall subgroup L of G contained in D. In general L can be arbitrary. The main result of this work, however, is to show that the commutator of L with a complement in G is nilpotent.