Algebra & Number Theory

Regular permutation groups of order mp and Hopf Galois structures

Timothy Kohl

Abstract

Let $Γ$ be a group of order $mp$ where $p$ is prime and $p>m$. We give a strategy to enumerate the regular subgroups of $Perm(Γ)$ normalized by the left representation $λ(Γ)$ of $Γ$. These regular subgroups are in one-to-one correspondence with the Hopf Galois structures on Galois field extensions $L∕K$ with $Γ= Gal(L∕K)$. We prove that every such regular subgroup is contained in the normalizer in $Perm(Γ)$ of the $p$-Sylow subgroup of $λ(Γ)$. 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 $p(p−1)$, where $p$ is a “safe prime”. These cases were previously studied by N. Byott and L. Childs, respectively.

Article information

Source
Algebra Number Theory, Volume 7, Number 9 (2013), 2203-2240.

Dates
Revised: 2 February 2013
Accepted: 11 March 2013
First available in Project Euclid: 20 December 2017

https://projecteuclid.org/euclid.ant/1513730091

Digital Object Identifier
doi:10.2140/ant.2013.7.2203

Mathematical Reviews number (MathSciNet)
MR3152012

Zentralblatt MATH identifier
1286.12002

Citation

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

References

• W. Burnside, Theory of groups of finite order, 2nd ed., Cambridge University Press, Cambridge, 1911. Reprinted by Dover in 1955.
• N. P. Byott, “Uniqueness of Hopf Galois structure for separable field extensions”, Comm. Algebra 24:10 (1996), 3217–3228.
• N. P. Byott, “Galois module theory and Kummer theory for Lubin–Tate formal groups”, pp. 55–67 in Algebraic number theory and Diophantine analysis (Graz, 1998), edited by F. Halter-Koch and R. F. Tichy, de Gruyter, Berlin, 2000.
• N. P. Byott, “Hopf–Galois structures on Galois field extensions of degree $pq$”, J. Pure Appl. Algebra 188:1-3 (2004), 45–57.
• S. U. Chase and M. E. Sweedler, Hopf algebras and Galois theory, Lecture Notes in Mathematics 97, Springer, Berlin, 1969.
• L. N. Childs, “On the Hopf Galois theory for separable field extensions”, Comm. Algebra 17:4 (1989), 809–825.
• L. N. Childs, “On Hopf Galois structures and complete groups”, New York J. Math. 9 (2003), 99–115.
• J. D. Dixon, “Maximal abelian subgroups of the symmetric groups”, Canad. J. Math. 23 (1971), 426–438.
• GAP Group, “GAP: groups, algorithms, and programming”, 2002, http://www.gap-system.org. Version 4.3.
• C. Greither and B. Pareigis, “Hopf Galois theory for separable field extensions”, J. Algebra 106:1 (1987), 239–258.
• I. M. Isaacs, Character theory of finite groups, Pure and Applied Mathematics 69, Academic Press, New York, 1976.
• T. Kohl, “Groups of order $4p$, twisted wreath products and Hopf–Galois theory”, J. Algebra 314:1 (2007), 42–74.
• M. Krasner and L. Kaloujnine, “Produit complet des groupes de permutations et problème d'extension de groupes, III”, Acta Sci. Math. Szeged 14 (1951), 69–82.
• J. A. Moody, Groups for undergraduates, World Scientific, River Edge, NJ, 1994.
• B. H. Neumann, “Twisted wreath products of groups”, Arch. Math. $($Basel$)$ 14 (1963), 1–6.
• O. Ore, “Theory of monomial groups”, Trans. Amer. Math. Soc. 51:1 (1942), 15–64.
• H. Wielandt, Permutationsgruppen, Mathematische Institut, Tübingen, 1955. Translated as Finite permutation groups, Academic Press, New York, 1964.