Let G be a nonempty set together with binary operation. We say G is a group under this operation if the following properties are satisfied :
Any pair of elements can be combined without going outside the set.
The operation is associative, that is (ab)c=a(bc) for all a,b,c in G.
There is an element e in G, such that ae=ea=a for all a in G.
For each element in G, there is an element b in G such that ab=ba=e.
Joseph A. Gallian book Page 51 no 1
Q: Give two reasons why the set of odd integers under addition is not a group.
A: The first, addition is not closed operation in the set of odd integers.For instance: 5+3=8. Eight is an even number.
The second, it doesn’t have identity. Zero (the identity of addition) is not element of the set of odd integers.