Subgroups of Additive Group of Integers/Examples/Multiples of 4

Example of Subgroup of Additive Group of Integers

Let $4 \Z$ denote the set of integers which are divisible by $4$.

Let $\struct {4 \Z, +}$ be the algebraic structure formed from $4 \Z$ with the operation of integer addition.

Then $\struct {4 \Z, +}$ is a subgroup of the additive group of integers $\struct {\Z, +}$.

Proof

The set of integers which are divisible by $4$ is the underlying set of the additive group of (integer) multiples of $4$.

Thus from Subgroups of Additive Group of Integers, $\struct {4 \Z, +}$ is a subgroup of $\struct {\Z, +}$.

$\blacksquare$

Sources

• 1967: George McCarty: Topology: An Introduction with Application to Topological Groups ... (previous) ... (next): Chapter $\text{II}$: Groups: Subgroups