Subgroups of Cartesian Product of Additive Group of Integers
Theorem
Let $\struct {\Z, +}$ denote the additive group of integers.
Let $m, n \in \Z_{> 0}$ be (strictly) positive integers.
Let $\struct {\Z \times \Z, +}$ denote the Cartesian product of $\struct {\Z, +}$ with itself.
The subgroups of $\struct {\Z \times \Z, +}$ are not all of the form:
- $\struct {m \Z, +} \times \struct {n \Z, +}$
where $\struct {m \Z, +}$ denotes the additive group of integer multiples of $m$.
Proof
Consider the map $\phi: \struct {m \Z, +} \times \struct {n \Z, +} \mapsto \struct {\Z, +} \times \struct {\Z, +}$ defined by:
- $\forall c, d \in \Z: \map \phi {m c, n d} = \tuple {c, d}$
which is a group isomorphism.
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Hence, $\struct {m \Z, +} \times \struct {n \Z, +}$ is a free abelian group of rank $2$.
Therefore, any subgroup generated by a singleton, for example, $\set {\tuple {x, 0}: x \in \Z}$ is a subgroup not in the form $\struct {m \Z, +} \times \struct {n \Z, +}$.
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$\blacksquare$
Sources
- 1971: Allan Clark: Elements of Abstract Algebra ... (previous) ... (next): Chapter $2$: Subgroups and Cosets: $\S 36 \delta$