Let ℝ∞ be the direct limit of the Euclidean spaces ℝn. Now the orthogonal group O(∞) acts on ℝn and the direct limit O(∞) of the groups O(∞) acts on ℝ∞. The infinite pin group Pin(∞) is an extension of ℤ2 by O(∞) and admits the following presentation: the generators are the unit vectors xf in ℝ∞ and the relations are

A length function, for a group, associates to an element x a real number |x| subject to certain axioms, including a cancellation axiom which embodies certain cancellation properties for elements of a free group. Integer valued length functions were introduced by Roger Lyndon [1] where, with a more restrictive set of axioms than ours, it is shown that a length function for a group is given by a restriction of the usual length function on some free product.

In this paper we continue our investigation of the topological filtration on the complex representation ring R(G) of a finite group, see [4] and [5]. To recall the basic definitions from (1): let

map a k-dimensional representation ζ to the (flat) vector bundle over the classifying space BG associated to the universal G-bundle by the G-structure on Ck. Then, if denotes the (2k − l)-skeleton of BG,

In this note I settle a question which arose out of my first paper under the above title (cf. [1]), where I considered the classgroup C(Z(Γ)) of the integral groupring Z(Γ) of a finite Abelian group Γ. This classgroup maps onto the classgroup C() of the maximal order of the rational groupring Q(Γ), and C() is the product of the ideal classgroups of the algebraic number fields which occur as components of Q(Γ) and is thus in a sense known. One is then interested in the kernel D(Z(Γ)) of C(Z(Γ)) → C() and in its order k(Γ). In [1] I proved that, for Γ a p-group, k(Γ) is a power of p. I also computed k(Γ) for small exponents. My computation used crucially the fact that, for the groups Γ considered, the groups of units of algebraic integers which occurred were finite, i.e. that the only number fields which turned up were Q and Q(n) with n4 = 1 or n6 = 1. The numerical results obtained led me to the question whether in fact k(Γ) tends to infinity with the order of Γ.