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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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