Universida de Federal de Minas Gerais
The wreath product construction is an essential tool in permutation group
theory. The wreath product of two permutation groups has two widely used
actions (namely the imprimitive action and the product action) and
understanding these actions is essential in reduction arguments for
permutation groups. In several of the classes identified by the O'Nan-Scott
Theorem for finite primitive and quasiprimitive groups, the members are
constructed using a wreath product. The twisted wreath product is a
generalization of the wreath product construction, and a number of
important primitive or quasiprimitive permutation groups can be described
as twisted wreath products.
In these lectures, we will review the most important properties of wreath
products and twisted wreath products. I will present necessary and/or
sufficient conditions for the primitivity or quasiprimitivity of such
wreath products. We will treat the following topics in more detail:
Wreath products, definitions, actions, imprimitive, primitive action, embedding theorems.
- Wreath products in the O'Nan-Scott theorem for primitive and
Twisted wreath products, the base group action, primitivity and
quasiprimitivity of twisted wreath products.
- Inclusions of (quasi)primitive groups into wreath products in
- Factorisations of groups and their application in the inclusion