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.