Suppose the set $G$ is an additive group of integers $(G,+)$. For a subset $H$ of the set $G$ to be a subgroup,
the subset $H$ must contain the identity element
the subset $H$ must be closed under the group binary operation
the subset $H$ must contain inverses of each of its elements
1) I generally have problem showing the identity element exists even though it is purported to be the easiest way to prove. Suppose I have shown that condition 2 and 3 holds, would adding the element of the subset $H$ to its inverse suffice to show that the identity element exists?
2) Would I have to show all of these condition holds or does showing just one condition suffice to show that the subset $H$ of a group $G$ is a subgroup.