I'm citing below the definition of and a theorem about cyclic groups, as it is written in my book (Algebra, by Thomas Hungerford):
Definition:
Let $ G $ be a group (notation is multiplicative in here.) For every $ a \in G, $ a cyclic group is: $ \langle a \rangle = \{a^n : n\in \mathbb{Z}\} $
and,
Theorem:
Every subgroup $ H $ of the additive group $ ( \mathbb{Z},+ ) $ is cyclic. Either $ H = \langle 0 \rangle, $ or $ H = \langle m \rangle, $ where $ m $ is the least positive integer in $ H $.
My question is;
I know that, and I would, express $ H $ as the union of all $ H_m = \{ m \mathbb{Z} : m \in H \} $.
The reason why I'm specifying this notation is that I don't understand the fact that $ m $ should be the least integer there (if I get it right of course.)
Thank you for giving advice.
Best Answer
You can use division with remainder to show that $H = m\mathbb{Z}$ for some $m$. For let $m$ be the smallest positive element of $H$. (If $H=\{0\}$, there is nothing to prove.) Then for any $k \in H$, we have
$$ k = q \cdot m + r \qquad \text{for some }q\text{, with } 0 \leq r < m.$$ Since both $k$ and $m$ lie in $H$ and $H$ is a subgroup, this means $r \in H$. But $m$ was the smallest positive element, so $r = 0$.