I asked this via MathSE, but haven't got any responces. Sorry for asking it here. Sorry.

We know that in the context of *abelian* groups, $p$-groups are called $p$-primary groups. I have a question about $p$-primary groups as follows. Derek J.S.Robinson, noted:

...the group $\mathbb Q/\mathbb Z$ is the direct sum of its primary components,

each of which is also divisible. Now the $p$-primary ...

when he was giving a basic concepts and ideas of Quasicyclic Groups in chapter 4 of his book *A course in the theory of groups*. In another reference, *An introduction to the theory of groups* by J.J.Rotman, we face to the following lemma in chapter 10:

Lemma 10.27. If $G$ and $H$ are divisible $p$-primary groups, then $G\cong H$ if and only if $G[p]\cong H[p]$.

I see that Robinson says any primary component is divisible naturally while in another view, Rotman is noting the groups which are divisible and $p$-primary.

Is being $p$-primitive leads us to being divisible? Or the whole groups are necessarily torsion?

Am I misunderstanding an important point here? Thanks for sharing your thoughts.