Maximal normal $\pi$-subgroups and torsion subgroups Let be $\mathbb{R}$ the real numbers and $\mathbb{Z}$ the integers. Let $G = \mathbb{R}/\mathbb{Z}$. Determine


*

*$O_\pi(G)$ - the normal maximal $\pi$-group of $G$,

*$T(G)$ - torsion subgroup.

 A: Try using the definitions. We don't use anything special. At each step, we try to make the definition less fancy; we try to write things down in a way we understood earlier in life.
An element of $G=\mathbb{R}/\mathbb{Z}$ is a coset $r+\mathbb{Z}$. What does it mean for $r+\mathbb{Z}$ to be in $T(G)$?
It means $r+\mathbb{Z}$ has order $n$ for some positive integer $n \in \{1,2,\ldots\}$. Well what does that mean?
It means $n(r+\mathbb{Z}) = 0 + \mathbb{Z}$ for some $n \in \{1,2,\ldots\}$. What does this mean?
It means $nr-0 \in \mathbb{Z}$. What does that mean?
It means $nr=m$ for some $m \in \mathbb{Z}$. What does that mean?
It means $r=\frac{m}{n}$. What does that mean?
It means $r \in \mathbb{Q}$. What does that tell us?
Oh, $T(G) = \{ r + \mathbb{Z} : r \in \mathbb{Q} \} =\mathbb{Q}/\mathbb{Z}$.
The elements of order dividing $n$ in $\mathbb{R}/\mathbb{Z}$ are just the fractions $\tfrac{a}{n} + \mathbb{Z}$.
What elements live a $\pi$-group? Oh, the elements  $\tfrac{a}{n} + \mathbb{Z}$ where $n$ is a $\pi$-number.
