Suppose we have a finitely presented group $G$ with free presentation $$ R\hookrightarrow F \twoheadrightarrow G. $$ To this presentation we may associate an extension with abelian kernel $$ R/R' \hookrightarrow F/R' \twoheadrightarrow G. $$ Here $R'$ denotes the commutator subgroup of $R$, and the $G$-module $R/R'$ is sometimes known as the relation module associated to the presentation.

Under what conditions is the group $F/R'$ finitely presentable?

To give a concrete example, consider the free product of cyclic groups $C_2\ast C_3$ with presentation $\langle a,b \mid a^2,b^3 \rangle$.

Is there a nice finite presentation of $F/R'$ in this case?

I would also be interested in any pointers to places in the literature where calculations for particular presentations are carried out.


Let $F$ be a finitely generated non-abelian free group with a non-trivial normal subgroup $R \lhd F$. Suppose that $G:=F/R$ is finitely presented. Then the group $F/R'$ is finitely presented if and only if $G$ is finite. In particular, if $G=C_2*C_3$ then $F/R'$ is not finitely presented.

This is mentioned in the paper of Kulikova and Ol'shanskii (MR2523056 Kulikova, O. V.; Olʹshanskiĭ, A. Yu. "On the finite presentability of the group $F/[M,N]$." (Russian. Russian summary) Vestnik Moskov. Univ. Ser. I Mat. Mekh. 2006, no. 6, 19--21, 62; translation in Moscow Univ. Math. Bull. 61 (2006), no. 6, 20–23 (2007); http://arxiv.org/abs/math/0703604) and is attributed to Shmel'kin (MR0193131 Šmelʹkin, A. L. "Wreath products and varieties of groups." (Russian) Izv. Akad. Nauk SSSR Ser. Mat. 29 1965 149–170).

  • $\begingroup$ This answers all my questions! Thanks. $\endgroup$
    – Mark Grant
    Apr 9 '13 at 6:21

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.