Polynomial Ideals and Classes of Finite Groups

The object of this note is to discuss the properties of some polynomials (on a countable set of indeterminates) attached to any finite group which generalize the Eulerian functions of a group defined by P. Hall (1936, Quart. J. Math.7, 134–151). In particular, I will define some classes of finite groups associated to prime ideals of the polynomial ring, and I will show that each finite group has a unique largest quotient in such a class of groups. This work is a generalization of the notion of the b-group introduced in an earlier paper (Bouc, 1996, J. Algebra183, 664–736) through a systematic use of the polynomial formalism of Section 7.2.5 there. For the readerʹs convenience, however, this paper is self-contained and the proofs of the results already stated are included.