Question

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.

Answer 1

Robinson only assserts that for the specific group $\mathbb{Q}/\mathbb{Z}$ the $p$-primary componenents/subgroups are divisible; and this would remain true replacing $\mathbb{Q}/\mathbb{Z}$ by any other divisible group.

Rotman's lemma is very different. Here, you suppose $G$ is divisible and in addition that it is $p$-primary (or to use a different terminology a $p$-group). And, the same for $H$. Then some result on $G$ and $H$ is proved.

So, the $p$-primary subgroups of a divisible group are divisible. But, certainly not every $p$-primary group is divisible; just consider finite cycylic groups of prime power order, for example.

To answer your additional question: it is not necessary for a divisible group to be torsion (think of the rationals), but $p$-primary groups are always torsion essentially by definition.