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0.1. The general setting. The theory of polynomial invariants of finite groups is a meeting point of many branches of mathematics such as number theory, combinatorics, mathematical physics, topology and many more. The best-known example is that of the invariants of the symmetric group Sn acting on the polynomial ring in n variables. The ring of invariant polynomials (see below) is generated by the elementary symmetric polynomials, a fact which is frequently used, say, in Galois theory. In this paper we want to draw attention to a question concerning the mod-p-invariants of subgroups of GL(n,Z). This can be interpreted as looking at certain families of subgroups of GL(n,Fp), one for each prime. How do the degrees of generators of this algebra vary in dependence on p? More concretely, the situation is as follows. Let R be a commutative unital ring and P an ideal in R. For a subgroup G ⊆ GL(n,R) we will denote by (G mod P ) its image under the canonical map to GL(n,R/P ). Sometimes we will omit the brackets in this notation. From now on, all prime ideals under consideration will be non-zero. The group (G mod P ) then acts on the free n-dimensional standard module M for R/P and thereby induces an action on the symmetric algebra of the (dual) module of M . We will be interested in the case where R is the ring of integers of an algebraic number field, in which case every non-zero prime ideal is maximal and hence R/P is a field. The symmetric algebra is the polynomial ring in n variables over the field. If a finite group G acts by automorphisms on a polynomial ring K[X1, . . . . . . , Xn], K a field, then we can always choose so-called primary and secondary invariants. Primary invariants are a set of n invariant polynomials which are algebraically independent. They exist by Noether normalisation

Some families of finite groups and their rings of invariants