Kronecker classes, normal coverings and chief factors of groups

For a group $G$, a subgroup $U \leq G$ and a group $\mathrm{Inn}(G) \leq A \leq \mathrm{Aut}(G)$, we say that $U$ is an $A$-covering group of $G$ if $G = \bigcup_{a\in A}U^a$. A theorem of Jordan (1872) implies that if $G$ is a finite group, $A = \mathrm{Inn}(G)$ 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 (1988) 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 $|G:U| \leq f(|A:\mathrm{Inn}(G)|)$. A key piece of evidence for this conjecture is a theorem of Praeger (1994), 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 $|G:U|\leq g(|A:\mathrm{Inn}(G)|,c)$ where $c$ is the number of $A$-chief factors of~$G$. Unfortunately, the proof of this result contains an error. In this paper, using a different argument, we give a correct proof of this theorem.

💡 Research Summary

The paper addresses a problem at the intersection of group theory and field theory concerning “covering subgroups” of a finite group G. For a subgroup U ≤ G and a subgroup A of the automorphism group satisfying Inn(G) ≤ A ≤ Aut(G), one says that U is an A‑covering subgroup if the union of all its A‑conjugates equals G, i.e. G = ⋃_{a∈A} U^{a}. Jordan’s classical theorem (1872) shows that when A = Inn(G) a proper covering cannot exist, so U = G. Neumann and Praeger (1988) conjectured a far‑reaching generalisation: there should exist a function f such that for any finite G, any A‑covering U, the index |G : U| is bounded solely in terms of n = |A : Inn(G)|, i.e. |G : U| ≤ f(n).

Praeger (1994) provided partial evidence by introducing a two‑variable function g(n,c) where c is the number of A‑chief factors of G/U^{A} (the A‑chief length of the quotient). He claimed that |G : U| ≤ g(n,c) holds for all finite groups. However, a subtle mistake appears in the inductive step when G is a cyclic p‑group: the argument incorrectly assumes the inequality p ≤ |G : U| in the degenerate case U = G, which is false and leaves the bound unproved.

The authors of the present paper identify this error and supply a completely new proof that restores the validity of Praeger’s theorem. Their approach consists of two main parts.

First, they treat the special situation where U supplements a minimal A‑invariant subgroup L of G (i.e. G = U L and L has no proper non‑trivial A‑invariant subgroups). In this case they prove a stronger statement: there exists a single‑variable function f such that |G : U| ≤ f(n) for all n. The proof splits according to whether L is abelian or non‑abelian. If L is abelian, a simple counting argument using the size of the A‑orbit of U∩L gives |G : U| ≤ n. If L is non‑abelian, they write L ≅ T^{k} with T a non‑abelian simple group. Using Lemma 2.1 they show that U∩L decomposes as a direct product of diagonal subgroups across a partition of the coordinates. By analysing the action of A on the set of simple factors, they introduce three parameters: r (the number of A‑orbits on the factors), s (the size of each block), and m (the number of Aut(T)‑conjugacy classes in T). Careful combinatorial arguments establish the inequalities m ≤ r and s ≤ r, which together with a known bound |T| ≤ h(m) (where h is a non‑decreasing function guaranteed by Pyber’s result) yield the estimate |G : U| ≤ h(n) n^{2}. Defining f(n)=max{n, h(n) n^{2}} completes this part.

Second, they handle the general case by induction on |G|. Assuming U^{A}=1 (the case U^{A}≠1 is reduced to a smaller quotient), they select a minimal A‑invariant subgroup L. If G=U L, the first part gives the desired bound. Otherwise, they consider the quotient G/L, apply the induction hypothesis to obtain |G : U L| ≤ g(n,c−1). They then define the A‑core K=(U L)^{A} and note that K < G, so its A‑chief length is at most c−1. Using the bound on |A : K| derived from the previous step (which is at most (n·g(n,c−1))! ), they apply the induction hypothesis again to K, obtaining a bound for |K : U∩K|. Finally they combine the two bounds: \