The only examples I have encountered of infinite $p$-groups with trivial center employ non-elementary methods in their construction. For instance, Example 9.2.5 of Scott's Group Theory is a perfectly satisfactory example, but it requires the wreath product (which, though an invaluable group-theoretic tool, is not what I consider an "elementary method").
Does anyone know of an example (of an infinite $p$-group with trivial center) that can be constructed and proven to have the claimed properties in a way that is friendly to, say, students of a first course in group theory? Perhaps a large product of finite groups or an easy-to-describe matrix group?
(I also welcome arguments for the nonexistence of such an example!)
Best Answer
Here is a matrix example, hopefully correct and also hopefully sufficiently simple:
Consider the group $U$ of $\infty\times \infty$ upper unipotent matrices with entries in $\mathbb F_p$, with all but finitely many entries equal to zero; so they have the form $$\begin{pmatrix} 1 & a_{1 2} & a_{1 3} & \cdots \\\ 0 & 1 & a_{2 3} & \cdots \\\ \vdots & \vdots & \vdots \end{pmatrix},$$ with $a_{i,j} \in \mathbb F_p$ and all but finitely many $a_{i j } = 0$.
Note that $U$ admits a homomorphism onto the upper unipotent matrices $U_n$ in $GL_n(\mathbb F_p)$ for any $n$, given by forgetting the $a_{i,j}$ for $i$ or $j$ greater than $n$. If $z$ is in the centre of $U$, then its image lies in the centre of $U_n$ for each $n$. But the centre of $U_n$ has trivial image in $U_{n-1}$, and so the centre of $U$ actually has trivial image in each $U_n$. Since $U$ evidently embeds into the projective limit of the $U_n$, we see that $U$ has trivial centre. (Intuitively, we've made the nilpotency class infinite, and so pushed the centre away to infinity.)