TITLE: infinite abelian group where all elements have order 1, 2, or 4
QUESTION [16 upvotes]: Let $A$ be a (not necessarily finitely generated) abelian group where all elements have order 1, 2, or 4. Does it follow that $A$ can be written as a direct sum $(\bigoplus _\alpha \mathbb Z/4) \oplus (\bigoplus_\beta \mathbb Z/2)$?

REPLY [12 votes]: What I said in the comments is true, and goes by the name
Prüfer's First Theorem: An abelian $p$-group $G$ with bounded exponent (an integer $k$ such that $g^k=1$ for all $g\in G$) is a direct sum of cyclic subgroups.
The proof is by induction on $k=p^e$, the base case $e=1$ being the vector-space case.
For the inductive step, write $pG=\oplus_\alpha\langle g_\alpha\rangle$, and choose $h_\alpha$ in $G$ with $ph_\alpha=g_\alpha$. Then the $h_\alpha$ generate a subgroup of $G$ (call it $H$) that is a direct sum of $\langle h_\alpha\rangle$. Let $L$ be a subgroup of $G$, maximal with respect to having trivial intersection with $H$. Then $L$ is also a direct sum of cyclic subgroups (by the vector-space case), and you can show $G=H\oplus L$.
Reference: Fundamentals of the Theory of Groups, by Kargapolov and Merzljakov, $\S$10.