KRONECKER CLASSES, NORMAL COVERINGS AND CHIEF FACTORS OF GROUPS

For a group G, a subgroup U ≤ G and a group A such that Inn(G) ≤ A ≤ Aut(G), we say that U is an A-covering group of G if G = Ua∈A Ua. A theorem of Jordan (1872), implies that if G is a finite group, A = Inn(G) and U is an A-covering group of G, then U = G. Motivated by a question concerning Kronecker classes of field extensions, Neumann and Praeger (1990) conjectured that, more generally, there is an integer function f such that if G is a finite group and U is an A-covering subgroup of G, then |G : U| ≤ f(|A : Inn(G)|). A key piece of evidence for this conjecture is a theorem of Praeger [‘Kronecker classes of fields and covering subgroups of finite groups’, J. Aust. Math. Soc. 57 (1994), 17–34], which asserts that there is a two-variable integer function g such that if G is a finite group and U is an A-covering subgroup of G, then |G : U| ≤ g(|A : Inn(G)|, c), where c is the number of A-chief factors of G. Unfortunately, the proof of this theorem contains an error. In this paper, using a different argument, we give a correct proof of the theorem.