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Can a p-group, where $p$ is a prime, have a trivial center? I'm supposed to consider the restricted wreath product of $C_p \wr T_p$, where $C_p$ is the cyclic group of order $p$ and $T_p$ is the $p$-primary part of the additive group of rationals modulo $1$.

To be honest, I don't understand how a wreath product can be constructed on an infinite group (is $T_p$ infinite? If not, how so?). Furthermore, how do I show the center is trivial?

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