Non-existence of Hopf–Galois structures and bijective crossed homomorphisms

Abstract

By work of C. Greither and B. Pareigis as well as N. P. Byott, the enumeration of Hopf-Galois structures on a Galois extension of fields with Galois group G may be reduced to that of regular subgroups of Hol(N ) isomorphic to G as N ranges over all groups of order |G|, where Hol(−) denotes the holomorph. In this paper, we shall give a description of such subgroups of Hol(N ) in terms of bijective crossed homomorphisms G −→ N , and then use it to study two questions related to non-existence of Hopf-Galois structures.

Hopf–Galois structures on finite extensions with quasisimple Galois group

“…It remains to consider the groups . In [18, Theorem 1.6], the author has shown that if is the double cover of with , then for all groups of order . We shall extend this result and prove: Theorem If is a finite quasisimple group, then for all groups of order .…”

“…Recall that is said to be quasisimple if and is simple, where is the commutator subgroup and is the center of . In [18, Theorem 1.3], the author has already shown that: Theorem If is a finite quasisimple group, then .…”

Hopf-Galois structures on a Galois S-extension

“…In what follows, suppose that g is bijective, and we shall prove Proposition 2.1 (b) By Proposition 2.1 (a), the group N is not perfect, so it has a proper and maximal characteristic subgroup M containing [N, N ]. The quotient N/M is then abelian, and by the proof of the second statement of [10,Theorem 1.7], we know that A n = g −1 (M). This means that g restricts to a bijective map res(g) : A n −→ M; res(g)(σ) = g(σ).…”

“…Since M = Z(N ), the map (2.4) is injective, by [10,Proposition 3.5 (c)], for example. It follows that Out(N ) embeds into Out(N/M) ≃ Out(A n ) and so is abelian.…”

“…Hence, we conclude that f(A n ) lies in Inn(T ) m . Let us remark that the argument in[10, Lemma 4.8] uses[10, Proposition 3.4], which is a consequence of the classification of finite simple groups. As shown in[10, Lemma 4.1], the relation (2.2) implies that:• g restricts to a homomorphism ker(f) −→ T m , • g −1 (M) is a subgroup of S n of index [N : M].…”