Hopf–Galois structures on almost cyclic field extensions of 2-power degree

Abstract

Let L/K be a finite separable field extension, and let E be the normal closure of L/K. Let G = Gal(E/K) and G = Gal(E/L). We call L/K almost cyclic if G has a normal cyclic complement in G. This includes the case that L/K is a cyclic Galois extension or a radical extension. We give a method for counting Hopf-Galois structures on an almost cyclic extension L/K. We then count the Hopf-Galois structures on an almost cyclic extension of degree 2 n , n 3, and determine how many of them are almost classical. This is analogous to a result of T. Kohl [T. Kohl, Classification of the Hopf-Galois structures on prime power radical extensions, J. Algebra 207 (1998) 525-546] which counts the Hopf-Galois structures on a radical extension of odd prime-power degree. In contrast to the odd prime-power degree case, however, we find that an almost cyclic extension L/K of 2-power degree has Hopf-Galois structures for which the Hopf algebra acting on L is not commutative.

Skew braces and Hopf–Galois structures of Heisenberg type

“…The second summation in l 1 arises by moving the τ a 2 v 2 j terms to gather them in one place using the relation τ σ = ρστ . Note, here l 1 and l 2 are divisible by r for r > 3 a prime number, so we find (vα a 1 1 α a 2 2 α a 3 3 ) p = 1 (10) for every v ∈ M 1 since p > 3. Note further that in (9), when a 2 = 0, we have…”

Mutually Normalizing Regular Permutation Groups and Zappa-Szep Extensions of the Holomorph

“…However, it turns out that the enumeration is nearly identical. In [6], Byott determined S(G) which we give here. let δ s ∈ Aut(G) be that automorphism such that δ s σδ −1 s = σ s .…”

“…As such Q(D 3 ) = H(D 3 ), and that NHol(G) = QHol(G) ∼ = (S 3 × S 3 ) ⋊ C 2 where the C 2 component is that element conjugating λ(D 3 ) to ρ(D 3 ), so that π(Q(G)) is also this same subgroup of order 2. Indeed, if we embed D 3 as S 3 into S 6 then we have λ(S 3 ) = (1, 3)(2, 5) (4,6), (1,4,5)(2, 6, 3) ρ(S 3 ) = (1, 2)(3, 5) (4,6), (1,4,5)(2, 3, 6) QHol(S 3 ) = Hol(S 3 ) τ for any τ ∈ { (1,4), (1,5), (2,3), (2,6), (3,6), (4,5)} so that all π(Q(S 3 )) are isomorphic, which is unsurprising given that Q = H and QHol(S 3 ) = NHol(S 3 ) so that any π(Q(G)) would be isomorphic to T (S 3 ). For D n = x, t | x n = 1, t 2 = 1, xt = tx −1 in general, in [15], we have a complete enumeration of R(D n ) = R(D n , [D n ]).…”

Minimal Hopf-Galois Structures on Separable Field Extensions

“…(b) Assume that K/k is a cyclic extension of degree 2 n for n ≥ 3. It is shown in [3] that K/k admits 3 • 2 n−2 Hopf Galois structures. Among them 2 n−2 of cyclic type, 2 n−2 of dihedral type and 2 n−2 of generalized quaternion type.…”