This is Exercise 4.3.8 of Robinson's "A Course in the Theory of Groups (Second Edition)". According to Approach0, it is new to MSE.
(NB: Modules aren't covered in Robinson's book yet.)
The Details:
Let $p$ be prime. A $p$-group is a group all of whose elements have a $p$ power order.
On page 12 of Robinson's book,
A torsion group [. . .] is a group all of whose elements have finite order.
On page 94, ibid.,
An element $g$ of an abelian group $G$ is said to be divisible in $G$ by a positive integer $m$ if $g=mg_1$ for some $g_1$ in $G$. [. . .]
An abelian group $G$ is said to be divisible of each element is divisible by every positive integer.
On page 106, ibid.,
A subgroup $H$ of an abelian group $G$ is called pure if
$$nG\cap H=nH$$
for all integers $n\ge 0$.
On page 107, ibid.,
Let $G$ be an abelian torsion group. A subgroup $B$ is called a basic subgroup if it is pure in $G$, it is the direct sum of cyclic groups, and $G/B$ is divisible.
On page 108, ibid.,
An additively written group is called bounded if its elements have boundedly finite orders: [. . .] multiplicative groups with this property are said to have finite exponent.
The Question:
An abelian $p$-group has a bounded basic subgroup if and only if it is the direct sum of a divisible group and a bounded group.
Thoughts:
Let $G$ be an abelian $p$-group.
There's a lot going on here. Divisible groups familiar to me:
- An abelian, characteristically simple group is divisible (supposedly).
- The pure subgroups of a divisible abelian group are just the direct summands.
- Show that a group, given by a presentation, is countable & reduced with a nontrivial element of infinite height
Of course, so are abelian groups, $p$-groups, cyclic groups, bounded groups (in their multiplicative form), pure subgroups, torsion groups, and direct sums.
I think $G$ has to be infinite.
It might be easier to consider when $G$ is finitely generated, since then we could use the fundamental theorem of finitely generated abelian groups to get $G$ as a direct sum of some groups; I'm not sure how to take it from there. However, Theorem 4.2.9, page 103, ibid., states,
A finitely generated abelian group $G$ is finite if it is a torsion group.
This is a question I think I could answer myself. However, I'm coming towards the end of my degree so I don't have enough time to spend on reading textbooks thoroughly - but I think I would lose momentum in Robinson's book if I left it any longer. The kind of answer I'm hoping for is a full solution.
Please help :)