Finding an isomorhpism $A_n \cong \mathbb{Z}/(m,n)\mathbb{Z}$

Question

Let me define $A_n = \{ a \in \mathbb{Z}/m\mathbb{Z} : an = 0 \}$. Could anyone give me a hint as to how to construct an explicit bijection between $A_n$ and $\mathbb{Z}/(n,m)\mathbb{Z}$. I really have no idea how to go about this. Any hint is highly appreciated!

Answer 1

Then you simply need to exhibit a homomorphism $\phi:\mathbb{Z}\rightarrow A_n$ for which $\ker\phi = (m,n)\mathbb{Z}.$ Let $(n,m) = d$ with $n = n_1d$ and $m = m_1d$. Then you can easily see that $A_n = \{a\in\mathbb{Z}/m\mathbb{Z} :an_1=0 \,\text{mod}\, m_1 \}=\{0,m_1,2m_1, ...(d-1)m_1\}=m_1(\mathbb{Z}/d\mathbb{Z}).$ Thus, simply let $\phi(k) = m_1(k\mod{d}).$ This is clearly a homomorphism and $\phi(k) = 0$ if and only if $k\in d\mathbb{Z} = (m,n)\mathbb{Z}$ and the first isomorphism theorem gives the desired isomorphism.