1
$\begingroup$

In page 43 of Kenneth S.Brown's book "Cohomology of Groups", GTM 87, we have a proposition:

If $G=F(S)/R$ then there is an exact sequence $0\to R_{ab}\overset{\theta}{\to} \mathbb{Z}G^{(S)}\overset{\epsilon}{\to}G\to 0$ of $G$-modules, where $\mathbb{Z}G^{(S)}$ is free with basis $(e_s)_{s\in S}$ and $\partial e_s=\bar{s}-1$.

Here according to page 45 Exercise 3(d) on this book, the above proposition implies that we have a partial free resolution

$$\mathbb{Z}G^{(T)}\overset{\partial_2}{\to} \mathbb{Z}G^{(S)}\overset{\partial_1}{\to}\mathbb{Z}G\overset{\epsilon}{\to}G\to 0$$

such that the matrix of $\partial_2$ is the "Jacobian matrix" $\overset{-}{(\partial t/\partial s)}_{t\in T, s\in S}$, which is related to the Fox calculs.

My question is:

In the presentation of $G$ above, could both the generators set $S$ and the relation set $T$ be infinite?

I also want to know in this book, does it always alow that the generator sets and relation sets be infinite in a presentation of a group $G$?

$\endgroup$
  • 1
    $\begingroup$ Here and in most of the book, there are no finiteness assumptions. The exception is Chapters VIII and IX. $\endgroup$ – Misha Apr 12 '13 at 22:49

Your Answer

By clicking "Post Your Answer", you acknowledge that you have read our updated terms of service, privacy policy and cookie policy, and that your continued use of the website is subject to these policies.

Browse other questions tagged or ask your own question.