# Three Notes on Abelian p-Groups

## Three Notes on Abelian p-Groups

### Abstract

Firstly, we prove that there is a separable p ω+1-projective abelian p-group G such that G/K is not a direct sum of countable groups for any countable pure subgroup K ≤ G. This parallels to a result from Math. Scand. (2007) achieved by Danchev-Keef. Secondly, we show the surprising fact that every abelian p-group is weakly socle-regular, which contrasts results pertaining to socle-regular abelian pgroups established in Archiv der Math. (2009) by Danchev-Goldsmith. And thirdly, we obtain that an arbitrary abelian p-group with finite Ulm Kaplansky invariants is L-co-Hopfian if, and only if, it is co-Hopfian. This supplies a recent result due to Chekhlov-Danchev in Internat. J. Algebra & Computat. (2017).

**Eurasian Bulletin of Mathematics**, [S.l.], v. 2, n. 1, p. 1-3, apr. 2019. Available at: <http://www.ebmmath.com/index.php/EBM/article/view/24>. Date accessed: 24 july 2019.