Subgroups of 3-Factor Direct Products

Extending Goursat’s Lemma we investigate the structure of subdirect products of 3-factor direct products. We construct several examples and then provide a structure theorem showing that every such group is essentially obtained by a combination of the examples. The central observation in this structure theorem is that the dependencies among the group elements in the subdirect product that involve all three factors are of Abelian nature. In the spirit of Goursat’s Lemma, for two special cases, we derive correspondence theorems between data obtained from the subgroup lattices of the three factors (as well as isomorphisms between arising factor groups) and the subdirect products. Using our results we derive an explicit formula to count the number of subdirect products of the direct product of three symmetric groups.

Characteristic subgroup lattices and Hopf–Galois structures

The Hopf–Galois structures on normal field extensions [Formula: see text] with [Formula: see text] are in one-to-one correspondence with the set of regular subgroups [Formula: see text] of [Formula: see text], the group of permutations of [Formula: see text] as a set, that are normalized by the left regular representation [Formula: see text]. Each such [Formula: see text] corresponds to a Hopf algebra [Formula: see text] that acts on [Formula: see text]. Such regular subgroups need not be isomorphic to [Formula: see text] but must have the same order. One can divide all such [Formula: see text] into collections [Formula: see text], where [Formula: see text] is the set of those regular [Formula: see text] normalized by [Formula: see text] and isomorphic to a given abstract group [Formula: see text], where [Formula: see text]. There exists an injective correspondence between the characteristic subgroups of a given [Formula: see text] and the set of subgroups of [Formula: see text] stemming from the Galois correspondence between sub-Hopf algebras of [Formula: see text] and intermediate fields [Formula: see text], where [Formula: see text]. We utilize this correspondence to show that for certain pairs [Formula: see text], the collection [Formula: see text] must be empty. This also shows that for these [Formula: see text], there do not exist skew braces with additive group isomorphic to [Formula: see text] and circle group isomorphic to [Formula: see text].

OVERGROUPS OF WEAK SECOND MAXIMAL SUBGROUPS

A subgroup $H$ is called a weak second maximal subgroup of $G$ if $H$ is a maximal subgroup of a maximal subgroup of $G$. Let $m(G,H)$ denote the number of maximal subgroups of $G$ containing $H$. We prove that $m(G,H)-1$ divides the index of some maximal subgroup of $G$ when $H$ is a weak second maximal subgroup of $G$. This partially answers a question of Flavell [‘Overgroups of second maximal subgroups’, Arch. Math.64(4) (1995), 277–282] and extends a result of Pálfy and Pudlák [‘Congruence lattices of finite algebras and intervals in subgroup lattices of finite groups’, Algebra Universalis11(1) (1980), 22–27].