I've been reading this article (http://arxiv.org/PS_cache/arxiv/pdf/1109/1109.0024v2.pdf, page 7, paragraph 2) about a generalized Goursat lemma and in the article the author determines the cyclic subgroups of $A \times B$. In the proof he makes the following statement (sort of):
Suppose we are given finite cyclic groups $\overline{G}_1, \overline{G}_2$ of orders $m=m_1d$ and $n=n_1d$, where $d=gcd(m, n)$. Let $G_1, G_2$ be subgroups of $\overline{G}_1, \overline{G}_2$ of coprime orders $m_1, n_1$. Suppose there exists an isomorphism $\theta: \overline{G}_1/G_1\rightarrow \overline{G}_2/G_2$. If $\alpha$ is a generator of $\overline{G}_1$ and $\beta G_2=\theta(\alpha G_1)$, then both $\alpha G_1$ and $\beta G_2$ are of order $d$, whence $\beta$ is of order $d|G_2|=dn_1=n$.
Why must $\beta$ be of order $n$?? All I can prove is that the order of $\beta$ is of the form $d\cdot r$, where $r|n_1$.