Hopf-Galois structures on extensions of degree $p^{2} q$ and skew braces of order $p^{2} q$: the elementary abelian Sylow $p$-subgroup case
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arXiv
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2023
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| author | Campedel, E. Caranti, A. Del Corso, I. |
| author_facet | Campedel, E. Caranti, A. Del Corso, I. |
| contents | Let $p, q$ be distinct primes, with $p > 2$. In a previous paper we classified the Hopf-Galois structures on Galois extensions of degree $p^{2} q$, when the Sylow $p$-subgroups of the Galois group are cyclic. This is equivalent to classifying the skew braces of order $p^2q$, for which the Sylow $p$-subgroups of the multiplicative group is cyclic. In this paper we complete the classification by dealing with the case when the Sylow $p$-subgroups of the Galois group are elementary abelian.
According to Greither and Pareigis, and Byott, we will do this by classifying the regular subgroups of the holomorphs of the groups $(G, \cdot)$ of order $p^{2} q$, in the case when the Sylow $p$-subgroups of $G$ are elementary abelian.
We rely on the use of certain gamma functions $γ:G\to \operatorname{Aut}(G)$. These functions are in one-to-one correspondence with the regular subgroups of the holomorph of $G$, and are characterised by the functional equation $γ(g^{γ(h)} \cdot h) = γ(g) γ(h)$, for $g, h \in G$. We develop methods to deal with these functions, with the aim of making their enumeration easier and more conceptual. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2303_13387 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Hopf-Galois structures on extensions of degree $p^{2} q$ and skew braces of order $p^{2} q$: the elementary abelian Sylow $p$-subgroup case Campedel, E. Caranti, A. Del Corso, I. Rings and Algebras Group Theory Number Theory 12F10 16W30 20B35 20D45 Let $p, q$ be distinct primes, with $p > 2$. In a previous paper we classified the Hopf-Galois structures on Galois extensions of degree $p^{2} q$, when the Sylow $p$-subgroups of the Galois group are cyclic. This is equivalent to classifying the skew braces of order $p^2q$, for which the Sylow $p$-subgroups of the multiplicative group is cyclic. In this paper we complete the classification by dealing with the case when the Sylow $p$-subgroups of the Galois group are elementary abelian. According to Greither and Pareigis, and Byott, we will do this by classifying the regular subgroups of the holomorphs of the groups $(G, \cdot)$ of order $p^{2} q$, in the case when the Sylow $p$-subgroups of $G$ are elementary abelian. We rely on the use of certain gamma functions $γ:G\to \operatorname{Aut}(G)$. These functions are in one-to-one correspondence with the regular subgroups of the holomorph of $G$, and are characterised by the functional equation $γ(g^{γ(h)} \cdot h) = γ(g) γ(h)$, for $g, h \in G$. We develop methods to deal with these functions, with the aim of making their enumeration easier and more conceptual. |
| title | Hopf-Galois structures on extensions of degree $p^{2} q$ and skew braces of order $p^{2} q$: the elementary abelian Sylow $p$-subgroup case |
| topic | Rings and Algebras Group Theory Number Theory 12F10 16W30 20B35 20D45 |
| url | https://arxiv.org/abs/2303.13387 |