Augmentation modules for affine groups (Q2710560)

In 1991, \textit{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 \textit{D. R. Farkas} and \textit{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.NEWLINENEWLINENEWLINEThe 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 [\textit{D. S. Passman}, Trans. Am. Math. Soc. 354, No. 8, 3379-3408 (2002; Zbl 0998.16017); \textit{D. S. Passman} and \textit{A. E. Zalesskij}, Proc. Am. Math. Soc. 130, No. 4, 939-949 (2002; Zbl 0992.16021); \textit{J. M. Osterburg}, \textit{D. S. Passman} and \textit{A. E. Zalesskij}, ibid. 130, No. 4, 951-957 (2002; Zbl 0992.16022)].