Transitive simple subgroups of wreath products in product action

Trans. Amer. Math. Soc.

2007

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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.

Section: With Gmentioning

confidence: 99%

Section: With Gmentioning

confidence: 99%

Section: Intransitive Homogeneous Cartesian Decompositionsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

Intransitive Cartesian decompositions preserved by innately transitive permutation groups

Baddeley¹,

Trans. Amer. Math. Soc.

2007

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Three types of inclusions of innately transitive permutation groups into wreath products in product action

2007

Isr. J. Math.

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Abstract. A permutation group is innately transitive if it has a transitive minimal normal subgroup, and this subgroup is called a plinth. In this paper we study three special types of inclusions of innately transitive permutation groups in wreath products in product action. This is achieved by studying the natural Cartesian decomposition of the underlying set that correspond to the product action of a wreath product. Previously we identified six classes of Cartesian decompositions that can be acted upon transitively by an innately transitive group with a non-abelian plinth. The inclusions studied in this paper correspond to three of the six classes. We find that in each case the isomorphism type of the acting group is restricted, and some interesting combinatorial structures are left invariant. We also show how to construct examples of inclusions for each type.

Group factorisations, uniform automorphisms, and permutation groups of simple diagonal type

2018

Isr. J. Math.

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We present a new proof, which is independent of the finite simple group classification and applies also to infinite groups, that quasiprimitive permutation groups of simple diagonal type cannot be embedded into wreath products in product action. The proof uses several deep results that concern factorisations of direct products involving subdirect subgroups. We find that such factorisations are controlled by the existence of uniform automorphisms.