Now, as Varun Tripathi mentioned closure is not satisfied by the set of odd integers. This is enough to conclude that the set is not a group under addition. But we can go further ahead and say that the identity of integers under addition, 0, does not belong to the set of odd integers as 0 is even. read more
The set of odd number is a subset of the integers; it is not a subgroup because it is not closed; for [math]a\in 2\mathbb{Z}+1[/math] (the odd integers), [math]a+a=2a[/math]; [math]2a[/math] is divisable by [math]2[/math],meaning [math]2a[/math] is even, so the subset is not closed under addition. read more
Because the proposed identity element and any other member of the group are both odd integers, and the sum of every two odd integers is even, the sum of the identity element and any odd integer is not even in the proposed group let alone equal to the odd integer. Remember, 0 is NOT an odd integer. read more