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:
- there exists a group homomorphism $\psi: \struct {\Z_m, +_m} \to \struct {\Z_n, +_n}$
- $m \divides n$
where $\divides$ denotes divisibility.
Proof
An example of the use of Third Isomorphism Theorem/Groups/Corollary.
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Sources
- 1967: George McCarty: Topology: An Introduction with Application to Topological Groups ... (previous) ... (next): Chapter $\text{II}$: Groups: Factoring Morphisms