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Some infinite permutation modules. (English) Zbl 0719.20002
Let G be a group acting on a set $$\Omega$$, F a field. Then the F- vectorspace with basis $$\Omega$$ is an FG-module in a natural way, the permutation module $$F\Omega$$ of (G,$$\Omega$$) over F. In the paper the submodule structure of $$F\Omega$$ is studied for several interesting classes of infinite transitive permutation groups (G,$$\Omega$$). The results include criteria for “almost irreducibility” (Theorem 2.1), indecomposability (Theorem 2.3), ascending chain condition (Theorem 2.4) and a discussion of important examples, mainly the infinite cyclic group, infinite symmetric groups acting on k-sets and the one-dimensional affine group over the rationals.

##### MSC:
 20C07 Group rings of infinite groups and their modules (group-theoretic aspects) 20B07 General theory for infinite permutation groups 20C32 Representations of infinite symmetric groups 20B22 Multiply transitive infinite groups
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