Let p,q be distinct primes, with p>2. We classify the Hopf-Galois structures on Galois extensions of degree p2q, such that the Sylow p-subgroups of the Galois group are cyclic. This we do, according to Greither and Pareigis, and Byott, by classifying the regular subgroups of the holomorphs of the groups (G,⋅) of order p2q, in the case when the Sylow p-subgroups of G are cyclic. This is equivalent to classifying the skew braces (G,⋅,∘). Furthermore, we prove that if G and Γ are groups of order p2q with non-isomorphic Sylow p-subgroups, then there are no regular subgroups of the holomorph of G which are isomorphic to Γ. Equivalently, a Galois extension with Galois group Γ has no Hopf-Galois structures of type G. Our method relies on the alternate brace operation ∘ on G, which we use mainly indirectly, that is, in terms of the functions γ:G→Aut(G) defined by g↦(x↦(x∘g)⋅g−1). 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(gγ(h)⋅h)=γ(g)γ(h), for g,h∈G. We develop methods to deal with these functions, with the aim of making their enumeration easier, and more conceptual.
Hopf-Galois structures on extensions of degree p2q and skew braces of order p2q: The cyclic Sylow p-subgroup case / Campedel, E.; Caranti, A.; Del Corso, I.. - In: JOURNAL OF ALGEBRA. - ISSN 0021-8693. - STAMPA. - 556:(2020), pp. 1165-1210. [10.1016/j.jalgebra.2020.04.009]
Hopf-Galois structures on extensions of degree p2q and skew braces of order p2q: The cyclic Sylow p-subgroup case
Caranti A.;
2020-01-01
Abstract
Let p,q be distinct primes, with p>2. We classify the Hopf-Galois structures on Galois extensions of degree p2q, such that the Sylow p-subgroups of the Galois group are cyclic. This we do, according to Greither and Pareigis, and Byott, by classifying the regular subgroups of the holomorphs of the groups (G,⋅) of order p2q, in the case when the Sylow p-subgroups of G are cyclic. This is equivalent to classifying the skew braces (G,⋅,∘). Furthermore, we prove that if G and Γ are groups of order p2q with non-isomorphic Sylow p-subgroups, then there are no regular subgroups of the holomorph of G which are isomorphic to Γ. Equivalently, a Galois extension with Galois group Γ has no Hopf-Galois structures of type G. Our method relies on the alternate brace operation ∘ on G, which we use mainly indirectly, that is, in terms of the functions γ:G→Aut(G) defined by g↦(x↦(x∘g)⋅g−1). 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(gγ(h)⋅h)=γ(g)γ(h), for g,h∈G. We develop methods to deal with these functions, with the aim of making their enumeration easier, and more conceptual.File | Dimensione | Formato | |
---|---|---|---|
p2q-cyclic_case_arXiv_v4.pdf
Open Access dal 16/08/2022
Descrizione: Post-print usato per aggiornare arXiv
Tipologia:
Post-print referato (Refereed author’s manuscript)
Licenza:
Creative commons
Dimensione
455.24 kB
Formato
Adobe PDF
|
455.24 kB | Adobe PDF | Visualizza/Apri |
1-s2.0-S0021869320301770-main.pdf
Solo gestori archivio
Tipologia:
Versione editoriale (Publisher’s layout)
Licenza:
Tutti i diritti riservati (All rights reserved)
Dimensione
698.14 kB
Formato
Adobe PDF
|
698.14 kB | Adobe PDF | Visualizza/Apri |
I documenti in IRIS sono protetti da copyright e tutti i diritti sono riservati, salvo diversa indicazione