[Math] Is the center of a p-group non-trivial

Question

Is it true that if $G$ is a $p$-group, where $p$ is a prime, then center of $G$ is non-trivial?

I know for finite $p$-group center of $G$ is non-trivial (easy to prove using class equation for group). But I am not sure about infinite $p$-group. I have no idea how to approach this problem. I know Prufer group is an example of infinite $p$-group. Is the center of Prufer group is trivial?

Any help would be appreciated…

Answer 1

The center of an infinite $p$-group can be trivial. From Rotman's An Introduction to the Theory of Groups, p. 115:

... there is an example of McLain (1954) of an infinite $p$-group $G$ with $Z(G)=1$, with $G'=G$ (so that $G$ is not even solvable), and with no characteristic subgroups other than $G$ and $1$.