Does $\mathbb Z/p\mathbb Z$ a free abelian pro-p group?

Question

As far as I know, $\mathbb Z/p\mathbb Z$ ($p$ is prime) is cyclic and so it's abelian.
It is obviously a p-group, hence it is pro-p.
And it is free, for its generator, $\langle1\rangle$, has no particular relation.

1) Is it correct to claim that $\mathbb Z/p\mathbb Z$ is a free abelian pro-p group?

2) In that context, is there a difference between the case $p=2$ and $p\neq2$?

Answer 1

It is neither "free" or "free abelian" since it has elements of finite order . You have the relation $1+\cdots+1=0$.