Algebra & Number Theory

Hopf–Galois structures arising from groups with unique subgroup of order $p$

Timothy Kohl

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Abstract

For Γ a group of order mp, where p is a prime with gcd(p,m) = 1, we consider the regular subgroups N Perm(Γ) that are normalized by λ(Γ), the left regular representation of Γ. These subgroups are in one-to-one correspondence with the Hopf–Galois structures on separable field extensions LK with Γ = Gal(LK). Elsewhere we showed that if p > m then all such N lie within the normalizer of the Sylow p-subgroup of λ(Γ). Here we show that one only need assume that all groups of a given order mp have a unique Sylow p-subgroup, and that p not be a divisor of the order of the automorphism groups of any group of order m. We thus extend the applicability of the program for computing these regular subgroups N and concordantly the corresponding Hopf–Galois structures on separable extensions of degree mp.

Article information

Source
Algebra Number Theory, Volume 10, Number 1 (2016), 37-59.

Dates
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
https://projecteuclid.org/euclid.ant/1510842464

Digital Object Identifier
doi:10.2140/ant.2016.10.37

Mathematical Reviews number (MathSciNet)
MR3463035

Zentralblatt MATH identifier
1341.20002

Subjects
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]

Keywords
Hopf–Galois extension regular subgroup

Citation

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


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References

  • J. Alonso, “Groups of square-free order, an algorithm”, Math. Comp. 30:135 (1976), 632–637.
  • 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-New York, 1969.
  • L. N. Childs, “On Hopf Galois structures and complete groups”, New York J. Math. 9 (2003), 99–115.
  • C. Greither and B. Pareigis, “Hopf Galois theory for separable field extensions”, J. Algebra 106:1 (1987), 239–258.
  • O. H ölder, “Die Gruppen mit quadratfreier Ordnungzahl”, Nachr. K önigl. Gesell. Wissenschaft. G öttingen Math.-Phys. Kl. (1895), 211–229.
  • T. Kohl, “Regular permutation groups of order $mp$ and Hopf Galois structures”, Algebra Number Theory 7:9 (2013), 2203–2240.
  • J. Pakianathan and K. Shankar, “Nilpotent numbers”, Amer. Math. Monthly 107:7 (2000), 631–634.