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Augmentation modules for affine groups. (English) Zbl 1005.20005
In 1991, D. M. Evans and the reviewer [Q. J. Math., Oxf. II. Ser. 42, No. 165, 15-26 (1991; Zbl 0719.20002)] considered the possible structure of permutation modules defined for some infinite permutation groups. As a result of the results in that paper the reviewer made a conjecture which this paper proves. Let \(F\) be a field and let \(G\) be the \(n\)-dimensional affine group \(\text{AGL}(n,F)\). Consider the permutation representation on the translation subgroup. Let \(k\) be a field and consider the permutation module over \(k\) for this action. This module always has at least one proper non-trivial \(kG\)-submodule, the augmentation module. For infinite fields the conjecture was that if \(k\) and \(F\) had different characteristics then the augmentation module is the only proper non-trivial \(kG\)-submodule. The case where \(F\) was the rational field was dealt with the Camina and Evans and when \(F\) is a finite extension of the rational field by D. R. Farkas and R. L. Snider [ibid. 45, No. 177, 29-42 (1994; Zbl 0802.20006)]. The situation for finite fields is well known and as a consequence it is easy to see that the conjecture holds for locally finite fields.
The proof is essentially ring theoretic. They consider the group algebra \(KA\) where \(A\) is the translation subgroup with the action of \(Q\), where \(Q=G/A\). Then the \(kG\)-submodules correspond to \(Q\)-invariant ideals. This enables the authors to use knowledge of the primes in the ring to be used. A number of recent papers have used these results [D. S. Passman, Trans. Am. Math. Soc. 354, No. 8, 3379-3408 (2002; Zbl 0998.16017); D. S. Passman and A. E. Zalesskij, Proc. Am. Math. Soc. 130, No. 4, 939-949 (2002; Zbl 0992.16021); J. M. Osterburg, D. S. Passman and A. E. Zalesskij, ibid. 130, No. 4, 951-957 (2002; Zbl 0992.16022)].

MSC:
20C07 Group rings of infinite groups and their modules (group-theoretic aspects)
20B07 General theory for infinite permutation groups
16S34 Group rings
16D25 Ideals in associative algebras
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