The term commutative group is used as a noun in sentences. A commutative group is a group that satisfies commutative law in mathematics. Commutative law states that we can swap numbers of problem when adding or multiplying.

Abelian meaning commutative. If the symmetry group of a square is commutative then it's an abelian group or else it's not.

abelian group

No. It is not a group.

Closure, an identity element, inverse elements, associative property, commutative property