The Uncountable Lattice of All Subfunctors of the Functor of Rational Representations

Abstract

In set theory the cardinality of the continuum $|{\mathbb R}|$ is the cardinal number of some interesting sets, like the Cantor set or the transcendental numbers. We will prove that the cardinal number of all subfunctors of the functor of rational representations $k \otimes_{{\mathbb Z}} R_{{\mathbb Q}}$, taking values on odd order groups over the field k of characteristic 2, is equal to $|{\mathbb R}|$. When the characteristic q > 0 of the field k is not necessarily even, we will present a formula giving the dimension of the evaluations S _C,k(G), of the simple functor S _C,k, at any group G of order prime to q and being associated to a suitable cyclic group C.