Center of a finitely generated group Could you give me an example (with proof) of a finitely generated group with a not finitely generated center?
 A: Here is an example I encountered recently (which unfortunately completely wrecked a proof I was working on at the time).
Start with a covering group $N$  of an infinite elementary abelian $p$-group for a prime $p$. This is not unique, but if we choose $p$ odd, then we can make it have exponent $p$. Then $N$ has a presentation
$\langle\  y_i, z_{jk}\ (i,j,k \in \mathbb{Z}, j<k) \mid [y_j,y_k] = z_{jk}\  (j<k),\  z_{jk} {\rm\ central},\  y_i^p=z_{jk}^p=1\  \rangle.$ 
This group has an automorphism of infinite order that maps $y_i \mapsto y_{i+1}$,
$z_{jk} \mapsto z_{j+1,k+1}$.
Take the semidirect product of $N$ with an infinite cyclic group $\langle x \rangle$ inducing this automorphism, and factor out the normal closure of the elements $z_{j,j+t}^{-1} z_{j+1,j+t+1}$ for all $t>0$. This yields a 2-generator group with presentation
$\langle\  y_1, x \mid y_1^p=1, [y_j,y_k] {\rm\ central\ for\ all\ } j<k\ \rangle,$
where $y_j$ is an abbreviation for $y_1^{x^j}$. Its centre is elementary abelian and generated by the infinite set $[y_1,y_{1+t}]$ for $t>0$.
A: V. N. Remeslennikov found a finitely presented group whose center is not finitely generated.  But I didn't check his construction.
Remeslennikov, V. N.
A finitely presented group whose center is not finitely generated. (Russian) 
Algebra i Logika 13 (1974), no. 4, 450–459, 488. 
English translation: Algebra and Logic 13 (1974), no. 4, 258–264 (1975)
