Subgroups of Additive Group of Integers/Examples/Even Integers

Example of Subgroup of Additive Group of Integers

Let $2 \Z$ denote the set of even integers.

Let $\struct {2 \Z, +}$ be the algebraic structure formed from $2 \Z$ with the operation of integer addition.

Then $\struct {2 \Z, +}$ is a subgroup of the additive group of integers $\struct {\Z, +}$.

Proof

The set of even integers is the underlying set of the additive group of (integer) multiples of $2$.

Thus from Subgroups of Additive Group of Integers, $\struct {2 \Z, +}$ 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: Exercise $\text{E ii}$
• 1996: John F. Humphreys: A Course in Group Theory ... (previous) ... (next): Chapter $4$: Subgroups: Exercise $1 \ \text{(a)}$
• 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): subgroup
• 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): subgroup