Let K be an extension of the field Qp of p-adic rationals, let OK be its ring of integers, and let G be a finite group. According to the classical Jordan–Zassenhaus Theorem, if K∕Qp is finite, every isomorphism class of KG-representation modules splits in a finite number of isomorphism classes of OKG-representation modules. We consider a p-group G of a given nilpotency class k>1 and the extension K∕Qp where K=Qp(ζp∞ ) obtained by adjoining all roots ζpi ,i=1,2,3,. of unity, and we use an explicit combinatorial construction of a faithful absolutely irreducible KG-module M to show that the number of OKG-isomorphism classes of OKG-representation modules, which are KG-equivalent to M, is infinite, and there are extra congruence properties for these OKG-modules.
ASJC Scopus subject areas
- Discrete Mathematics and Combinatorics