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 $G$?

  • 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

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