Prime power coverings of groups and field extensions

For a finite group A with normal subgroup G , a subgroup U of G is an A -prime-power-covering subgroup if U meets every A -conjugacy-class of elements of G of prime-power order. It is conjectured that | G : U | $|G:U|$ StartAbsoluteValue upper G colon upper U EndAbsoluteValue is bounded by some function of | A : G | $|A:G|$ StartAbsoluteValue upper A colon upper G EndAbsoluteValue . We prove the conjecture in the case that the action of G on the set of right cosets of U in G is innately transitive. This includes the special case where U is a maximal subgroup of G . It turns out that each prime-power-covering subgroup U for which this G -action is innately transitive must be a maximal subgroup of G . We derive the following number-theoretic consequence: if K is a finite Galois extension of a global field k and L is a finite innate extension of K such that K and L are weakly Kronecker equivalent over k , then the extension L / K $L/K$ upper L divided by upper K must be atomic of degree bounded in terms of [ K : k ] $[K:k]$ left bracket upper K colon k right bracket . Our proofs use both a recently proved bound on the order of a finite nonabelian simple group in terms of the number of its classes of elements of prime-power order (depending on the Finite Simple Group Classification) and also new Galois-theoretic concepts for field extensions.