Algebra & Number Theory
- Algebra Number Theory
- Volume 10, Number 1 (2016), 37-59.
Hopf–Galois structures arising from groups with unique subgroup of order $p$
For a group of order , where is a prime with , we consider the regular subgroups that are normalized by , the left regular representation of . These subgroups are in one-to-one correspondence with the Hopf–Galois structures on separable field extensions with . Elsewhere we showed that if then all such lie within the normalizer of the Sylow -subgroup of . Here we show that one only need assume that all groups of a given order have a unique Sylow -subgroup, and that not be a divisor of the order of the automorphism groups of any group of order . We thus extend the applicability of the program for computing these regular subgroups and concordantly the corresponding Hopf–Galois structures on separable extensions of degree .
Algebra Number Theory, Volume 10, Number 1 (2016), 37-59.
Received: 5 August 2014
Revised: 1 October 2015
Accepted: 27 November 2015
First available in Project Euclid: 16 November 2017
Permanent link to this document
Digital Object Identifier
Mathematical Reviews number (MathSciNet)
Zentralblatt MATH identifier
Primary: 20B35: Subgroups of symmetric groups
Secondary: 20D20: Sylow subgroups, Sylow properties, $\pi$-groups, $\pi$-structure 20D45: Automorphisms 16T05: Hopf algebras and their applications [See also 16S40, 57T05]
Kohl, Timothy. Hopf–Galois structures arising from groups with unique subgroup of order $p$. Algebra Number Theory 10 (2016), no. 1, 37--59. doi:10.2140/ant.2016.10.37. https://projecteuclid.org/euclid.ant/1510842464