Condition for Factoring of Quotient Mapping between Modulo Addition Groups

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Theorem

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

Let $\struct {\Z, +}$ denote the additive group of integers.

Let $\struct {\Z_m, +_m}$ and $\struct {\Z_n, +_n}$ denote the additive groups of integers modulo $m$ and $n$ respectively.

Let $f: \Z \to \Z_n$ be the quotient epimorphism from $\struct {\Z, +}$ to $\struct {\Z_n, +_n}$.

Let $q: \Z \to \Z_m$ be the quotient epimorphism from $\struct {\Z, +}$ to $\struct {\Z_m, +_m}$.


Then Then $N \subseteq K$


Then:

there exists a group homomorphism $\psi: \struct {\Z_m, +_m} \to \struct {\Z_n, +_n}$

if and only if

$m \divides n$

where $\divides$ denotes divisibility.


Proof

An example of the use of Third Isomorphism Theorem/Groups/Corollary 2.



Sources