Count of Distinct Homomorphisms between Additive Groups of Integers Modulo m

Theorem

Let $m, n \in \Z_{>0}$ be (strictly) positive integers.

Let $\struct {\Z_m, +}$ denote the additive group of integers modulo $m$.

The number of distinct homomorphisms $\phi: \struct {\Z_m, +} \to \struct {\Z_n, +}$ is $\gcd \set {m, n}$.

Proof

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$\Z_m$ is isomorphic to the quotient group $\Z / m\Z$.

By Universal Property of Quotient Group, to give a group homomorphism from $\Z_m$ to $\Z_n$ is equivalent to give a homomorphism $\varphi$ from $\Z$ to $\Z_n$ with kernel including the subgroup $m\Z \subset \Z$.

$\Z$ is generated by the element $1$.

By Homomorphism of Generated Group, $\varphi$ is determined by $\varphi(1)$.

The kernel condition means $\varphi(m) = m \varphi(1) = 0 \in \Z_n$.

Number of possible such $\varphi(1)$ is exactly $\gcd \set {m, n}$.

$\blacksquare$

Sources

• 1971: Allan Clark: Elements of Abstract Algebra ... (previous) ... (next): Chapter $2$: Group Homomorphism and Isomorphism: $\S 60 \zeta$