How determinate the torsion subgroup and the normal maximal $\pi$-subgroup of $\mathbb{R}/\mathbb{Z}$? Let $\mathbb{R}$ and $\mathbb{Z}$ the real numbers and the integers, respectively. Consider $G =\mathbb{R}/\mathbb{Z}$. Find
$O_{\pi}(G)$ - ie, the (unique) normal maximal $\pi$-subgroup of $G$;
$T(G)$ - ie, the torsion subgroup of $G$.
 A: If $G$ is a torsion abelian group it satisfies the $p$-group decomposition $G\cong\bigoplus G_p$, where each $G_p$ is the subgroup of all $p$-torsion elements (i.e. elts whose order is a power of $p$) of $G$, equivalently the union $\bigcup G[p^r]$ where $G[n]$ is defined as the subgroup of all $g\in G$ satisfying $g^n=e$.
If $G$ is a torsion abelian group and $\pi\subseteq{\cal P}$ a set of primes, a $\pi$-subgroup of $G$ is one in which every element has order which factors into primes only coming from $\pi$. Thus the maximal $\pi$-subgroup of any torsion abelian group $G$ is $\bigoplus_{p\in\pi}G_p$. Abelian automatically implies normal.
Note that $\bf R$ is an uncountable $\frak c$-dimensional $\bf Q$-vector space, so ${\bf R}\cong \bigoplus_{\frak c}{\bf Q}$. As $\bf Z$ is a subgroup of only one of the factors of $\bf Q$ in this decomposition, quotienting by $\bf Z$ only affects that factor, that is we have ${\bf R}/{\bf Z}\cong {\bf Q}/{\bf Z}\times\bigoplus_{\frak c}{\bf Q}$. All of the latter factors are nontorsion, so $T({\bf R/Z})\cong{\bf Q}/{\bf Z}$.
Finally, we have the Prüfer decomposition ${\bf Q}/{\bf Z}\cong \bigoplus_p {\bf Z}(p^\infty)$. The Prüfer $p$-groups are defined additively as ${\bf Z}[p^{-1}]/{\bf Z}$, or multiplicatively as the roots of unity in $\bf C$ with $p$-power order. This is in fact the $p$-torsion decopmosition of $G={\bf Q}/{\bf Z}$, so the maximal $\pi$-subgroup is $\bigoplus_{p\in\pi}{\bf Z}(p^\infty)$.
Tangent. An equivalent formulation is ${\bf Z}(p^\infty)\cong{\bf Q}_p/{\bf Z}_p$, where ${\bf Q}_p$ is the field of $p$-adic numbers with subring ${\bf Z}_p$ the $p$-adic integers. This allows the Prüfer decomposition to generalize to arbitrary global fields (quotienting a global field by its ring of integers decomposes as a direct sum of the local fields modulo their valuation subrings), in particular the function field case recovers the idea of partial fraction decomposition in algebraic language.
