Q:

$\kappa$-free abelian groups and inaccessible cardinals

I found the following unproved statements in Shelah and Mekler's article "On the consistency strength of "every stationary set reflects"".

Let $\kappa$ be the least cardinal such that there exists a $\kappa$-free abelian group which is not $\kappa^+$-free. By a well known
  argument, $\kappa$ is either the successor of a singular cardinal or an inaccessible
  cardinal. 
It is easy to see (and well known) that if every stationary set
  reflects in a regular cardinal then every $\kappa$-free abelian group is
  $\kappa^+$-free.

Can someone please give references for the above statements? In particulary I am deeply interested in the first one, but I couldn't find any introductory material. The wonderful book "Almost free modules" by Paul Eklof strangely doesn't mention the result (or I couldn't find it!).
Thank you in advance.

A:

The paper 

MR1029909 (91b:03090). Mekler, Alan H.; Shelah, Saharon. The consistency strength of "every stationary set reflects". Israel J. Math. 67 (1989), no. 3, 353–366,

that you mention in the question actually provides the relevant references and explains the key idea of the argument. Note first that $\kappa$ is assumed regular. 
They refer to 

MR0379694 (52 #599). Eklof, Paul C. On the existence of $\kappa$-free abelian groups. Proc. Amer. Math. Soc. 47 (1975), 65–72. 

In that paper, Eklof essentially shows that if $\kappa$ is regular and there is a stationary subset of $\kappa$ that does not reflect, then there is a $\kappa$-free not $\kappa^+$-free abelian group. This means that if we want the opposite, every stationary subset of $\kappa$ must reflect.
The point then is to note that $S^{\tau^+}_\tau$ is non-reflecting whenever $\tau$ is regular. Here $S^{\tau^+}_\tau$ is the set of ordinals smaller than $\tau^+$ of cofinality $\tau$. To see this, let $\alpha\in S^{\tau^+}_\tau$, fix a club subset $C$ of $\alpha$ of type $\tau$, and note that $C'$ cannot meet $S^{\tau^+}_\tau$ since no point of $C'$ has cofinality $\tau$. 
This leaves us with $\kappa$ being inaccessible or the successor of a singular, since all other regular cardinals are of the form $\kappa^+$ for $\kappa$ regular.

