Intransitive Cartesian decompositions preserved by innately transitive permutation groups

Abstract

Abstract. A permutation group is innately transitive if it has a transitive minimal normal subgroup, which is referred to as a plinth. We study the class of finite, innately transitive permutation groups that can be embedded into wreath products in product action. This investigation is carried out by observing that such a wreath product preserves a natural Cartesian decomposition of the underlying set. Previously we classified the possible embeddings in the case where the innately transitive group projects onto a transitive subgroup of the top group. In this article we prove that the transitivity assumption we made in the previous paper was not too restrictive. Indeed, the image of the projection into the top group can only be intransitive when the finite simple group that is involved in the plinth comes from a small list. Even then, the innately transitive group can have at most three orbits on an invariant Cartesian decomposition. A consequence of this result is that if G is an innately transitive subgroup of a wreath product in product action, then the natural projection of G into the top group has at most two orbits.

Embedding Permutation Groups Into Wreath Products in Product Action

“…https://doi.org/10.1017/S1446788712000110 132 C. E. Praeger and C. Schneider [6] (d) (ϑ, χ) is a permutational embedding of X on Π into the group Sym Ω 0 × Sym Ω 1 in its product action on Ω 0 × Ω 1 , and Xχ ≤ W 0 × W 1 .…”

“…2] or [8, Sections 1.10 and 4.3]) and finite quasiprimitive groups [13]. They have received special attention recently in the work of Aschbacher [1,2] aimed at studying intervals in subgroup lattices [3] (with Shareshian), and of the authors [4][5][6]14] action of the wreath product W = Sym Γ Sym ∆ is its natural action on the set Π = Func(∆, Γ) of functions from ∆ to Γ, described in Section 1.1. If ∆ = {1, .…”

“…In order to facilitate our discussion of subgroups of wreath products we invoke the language of Cartesian decompositions which was introduced by Baddeley and the authors [4] and was subsequently used to describe innately transitive subgroups of wreath products in product action [5,6,14]. Consider the set Π = Func(∆, Γ), and define, for each δ ∈ ∆, a partition Γ δ of Π as follows.…”

Quasiprimitive groups and blow-up decompositions

“…Lifting the restriction that the socle should be simple, we found that the variety of possible inclusions (of quasiprimitive groups in wreath products in product action) far outstripped anything we could have imagined. We developed a theory of such inclusions and many of the details of this theory are described in [3][4][5]18]. Our theory can be viewed as a generalisation of Baumeister's results [7], which are concerned with the subgroups G of S n S 2 such that G projects onto the top group S 2 and such that the projections of G on the two components of S n × S n are primitive and faithful.…”

“…k > 1 Transitive 6 types identified studied; further in [4,17,18] [5] k > 1 Intransitive Top group has two orbits; 5 types are identified product of smaller components. A Cartesian decomposition in which all the components have the same size is said to be homogeneous.…”