TITLE: Structure Theorem for abelian torsion groups that are not finitely generated?
QUESTION [44 upvotes]: I know about the structure theorem for finitely generated abelian groups.
I'm wondering whether there exists a similar structure theorem for abelian groups that are not finitely generated. In particular, I'm interested in torsion groups. Maybe having a finite exponent helps?

REPLY [14 votes]: To complement Arturo's excellent survey of the foundations of abelian p-groups:  A useful survey of the later (1960-2000) development of the theory of abelian p-groups is given in:
Hill, Paul. "The development of the theory of p-groups."
Rocky Mountain J. Math. 32 (2002), no. 4, 1135–1151.
MR1987598
DOI:10.1216/rmjm/1181070013.
I personally enjoyed the earlier theory a lot more (Kulakoff, Prüfer, Zippin, Ulm) which was gathered up into textbook form by Fuchs and Kaplansky.  However, lots of positive and negative results were achieved after then, so if you aren't daunted by the several hundred pages of the textbooks by Fuchs, Kaplansky, and Griffith, then the survey has a lot of important papers beyond the textbook level.