Informally, a subgroup is a group H contained within a bigger one, ⁠ G ⁠ : it has a subset of the elements of ⁠ G ⁠, with the same operation. Concretely, this means that the identity element of G must be contained in ⁠ H ⁠, and whenever h 1 and h 2 are both in ⁠ H ⁠, then so are h 1 ⋅ ⋅ h 2 and ⁠ h 1 − − 1 ⁠, so the elements of ⁠ H ⁠, equipped with the group operation on G restricted to ⁠ H ⁠, indeed form a group. In this case, the inclusion map H → → G is a homomorphism.