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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.

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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