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Simple augmentation modules. (English) Zbl 0802.20006
Let $$k$$ be a field, $$G$$ be a group, $$k[G]$$ be the group algebra of the group $$G$$ over the field $$k$$. If $$X$$ is a set, $$G$$ is a permutation group on $$X$$, then $$k[X]$$ is the $$k$$-vector space with $$X$$ as a basis, it becomes a $$k[G]$$-module by extending the action of $$G$$ on $$X$$ in the obvious way. Put $$\omega_ k(X) = \{\sum_{x \in X} \lambda_ x x\mid\sum_{x \in X} \lambda_ x = 0\}$$. The submodule $$\omega_ k(X)$$ is called the augmentation module.
The basic results here are Theorem 9. Let $$X$$ be an infinite set, $$G$$ be a permutation group on $$X$$ and $$\omega_ k(X)$$ is a simple $$k[G]$$- module. Then $$X \setminus \{y\}$$ has no finite $$\text{Stab}_ G(y)$$- orbits for any $$y \in X$$. Theorem 11. Suppose $$G$$ acts effectively on the infinite set $$X$$ and $$\omega_ k(X)$$ is a simple $$k[G]$$-module. Then $$\text{FC}\{G\} = \{x \in G\mid | G: C_ G(x)| \text{ is finite}\} = \langle 1\rangle$$. Theorem 13. Let $$G$$ be a group, $$A$$ be a nonidentity, normal, torsion free abelian subgroup of finite rank. Suppose that $$G$$ acts effectively on $$X$$. Then $$\omega_ k(X)$$ is a simple $$k[G]$$-module if and only if $$\text{char }k > 0$$ and no intermediate subgroup of $$A$$ has a finite $$G$$-orbit.

##### MSC:
 20C07 Group rings of infinite groups and their modules (group-theoretic aspects) 16S34 Group rings 20B07 General theory for infinite permutation groups 16D60 Simple and semisimple modules, primitive rings and ideals in associative algebras
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