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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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