Hopf-Galois structures of isomorphic-type on a non-abelian characteristically simple extension

Abstract

Let L/K be a finite Galois extension whose Galois group G is non-abelian and characteristically simple. Using tools from graph theory, we shall give a closed formula for the total number of Hopf-Galois structures on L/K with associated group isomorphic to G.

Rota-Baxter groups, skew left braces, and the Yang-Baxter equation

“…where G is non-abelian simple. Earlier [33], he showed that there are exactly e(G, G) = 2 n (n|Aut(G)| + 1) n−1 regular subgroups of Hol(G n ) which are isomorphic to G n . It is interesting to compare this value with 2 n (n+1) n−1 , the number of all RB-operators of weight 1 on the algebra k n [16].…”

Hopf–Galois structures on finite extensions with quasisimple Galois group

“…By (4.1), we know that |M | = |Z(G)| divides 48 or 36. Using the fact that Aut(M ) must be insolvable, we checked in Magma that M has SmallGroup ID equal to one of (8,5), (16,14), (24, 15), (48, 50), (48, 51), (48, 52), (4.8) and in particular m(G/Z(G)) is divisible by 8. Hence, we have G/Z(G) PSL 3 (4), and note that |PSL 3 (4)| = 20160.…”

“…Again, using the OuterOrder command, we computed in Magma that |Out(M )| = 168, 20160, 336, 120, 1344, 40320, respectively, when M has SmallGroup ID in (4.8). Moreover, the group M is abelian and there is no subgroup isomorphic to PSL 3 (4) in Aut(M ), when M has SmallGroup ID equal to (16,14),(48,52). We then deduce that G/Z(G) cannot embed into Out(M ).…”

“…If with , however, then by [6, Theorems 5 and 9], we have See [19, 20] for generalizations to other finite almost simple groups . If is a finite non‐abelian characteristically simple group which is not simple, then by [16], but as far as the author knows, there is no investigation yet on whether there exists such that in this case.…”

Hopf-Galois structures on finite extensions with quasisimple Galois group

“…It remains to consider the groups N ≃ G. In [16,Theorem 1.6], the author has shown that if G is the double cover of A n with n ≥ 5, then e(G, N ) = 0 for N of order n! with N ≃ G. We shall extend this result and prove: See [19,21] for generalizations to other finite almost simple groups G. If G is a finite non-abelian characteristically simple group which is not simple, then e(G, G) = 2 by [17], but as far as the author knows, there is no investigation yet on whether there exists N ≃ G such that e(G, N ) = 0.…”