The odd integers are not a subgroup of the integers

The odd integers are not a subgroup of the integers
Tag	`VFYVR2H4QS`
URI	`m2://VFYVR2H4QS`
Kind	example
Writing	W0 stub
MSC	20A05
Not an instance of	Subgroup (note: the sum of two odd integers is even.)
Uses	Integers, Subgroup

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Example

Let $G = (\mathbb{Z}, +)$ be the integers with addition, and let $O$ be the set of odd integers. The identity $0$ is not odd, so $0 \notin O$ . Also, $1 \in O$ and $1 + 1 = 2 \notin O$ . Thus $O$ is not closed under addition. Thus $O$ is not a subgroup of $G$ .

Remarks

$O$ is a coset of the even integers. A coset $g + H$ is a subgroup only when $g \in H$ .

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Group concept
Integers object
Subgroup definition
The odd integers are not a subgroup of the integers example

Sources

Web https://en.wikipedia.org/wiki/Subgroup, Subgroup tests
Web https://en.wikipedia.org/wiki/Quotient_group, Even and odd integers

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