Algebra & Number Theory
- Algebra Number Theory
- Volume 7, Number 9 (2013), 2203-2240.
Regular permutation groups of order mp and Hopf Galois structures
Let be a group of order where is prime and . We give a strategy to enumerate the regular subgroups of normalized by the left representation of . These regular subgroups are in one-to-one correspondence with the Hopf Galois structures on Galois field extensions with . We prove that every such regular subgroup is contained in the normalizer in of the -Sylow subgroup of . This normalizer has an affine representation that makes feasible the explicit determination of regular subgroups in many cases. We illustrate our approach with a number of examples, including the cases of groups whose order is the product of two distinct primes and groups of order , where is a “safe prime”. These cases were previously studied by N. Byott and L. Childs, respectively.
Algebra Number Theory, Volume 7, Number 9 (2013), 2203-2240.
Received: 8 September 2012
Revised: 2 February 2013
Accepted: 11 March 2013
First available in Project Euclid: 20 December 2017
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Kohl, Timothy. Regular permutation groups of order mp and Hopf Galois structures. Algebra Number Theory 7 (2013), no. 9, 2203--2240. doi:10.2140/ant.2013.7.2203. https://projecteuclid.org/euclid.ant/1513730091