DOI: 10.1017/fms.2026.10281 ISSN: 2050-5094
Prime power coverings of groups and field extensions
Michael Giudici, Luke Morgan, Cheryl E. Praeger Abstract
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.