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Could you give me an example (with proof) of a finitely generated group with a not finitely generated center?

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Abels gave an example of a finitely presented solvable group with non-finitely generated center (n example of a finitely presented solvable group. Homological group theory (Proc. Sympos., Durham, 1977), pp. 205–211, London Math. Soc. Lecture Note Ser., 36, Cambridge Univ. Press, Cambridge-New York, 1979.): the group of upper triangular matrices with coefficients in $\mathbb{Z}[1/p]$ ($p$ a prime), with $1$ in the top left and bottom right diagonal entries, and the other two entries positive units. I haven't checked the details, though. –  Arturo Magidin Feb 15 '12 at 21:09

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up vote 6 down vote accepted

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 $t>0$.

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why its center is elementary abelian? –  John Feb 17 '12 at 1:27
    
The relations $y_j^p=1$ together with the fact that $[y_j,y_k]=1$ imply that $[y_j,y_k]^p=1$. (The relations $z_{jk}^p=1$ in the presentation of $N$ are actually redundant.) –  Derek Holt Feb 17 '12 at 8:50

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