The coherence number of a finite group
The algebraic problem we will identify arose within algebraic topology in the course of the development of a transfer for homotopy theory. This transfer can be applied to the action of finite groups on topological spaces. More specifically, to each finite group G the theory associates, in a natural way, a distinguished element \(\theta_ G\) in a subgroup \(T_ G\) of a direct product of free groups. In order to apply the transfer, one is first required to compute the order of the image of \(\theta_ G\) in the abelianization, \(T_ G^{ab}\), of the subgroup \(T_ G\). It is this order, possibly infinite, which we have labelled as being the ``coherence number'' of the group G. As we will show, \(T_ G^{ab}\) is isomorphic to \({\mathbb{Z}}_{c(G)}\oplus {\mathbb{Z}}^{2| G| -1}\), so the group \(T_ G^{ab}\) is completely determined by the coherence number and order of G. As we will show, there are some delicate connections between groups and their subgroups and their respective coherence numbers. Fortunately it is true that the coherence number of a subgroup divides the coherence number of the group, but the situation for quotient groups is more subtle. The best that we have done so far is to show that the coherence number of the quotient of G by a normal subgroup H divides the product of the order of H and the coherence number of G. It is unknown in general whether the coherence number of a group is always finite. In general, it seems that it will be very large.
scientific article; zbMATH DE number 4147096