Here is the definition.
Let a be an element of the group G. If there exists a positive integer n such that an = e, then a is said to have Finite Order, and the smallest such positive integer is called the order of a, denoted by o(a). If there does not exist a positive integer n such that an = e, then a is said to have Infinite Order.
There is no way to prove that the given group (G,*) is a finite group or not. Because It does not talk about the sets. I have read this document. You can also read if you want to see that.
Now In the above question, The correct answer is C. The given group is an Abelien group. A group with Commutative property is called Abelian group. Because we will get (a * b)^2 = (a * a) * (b * b), only when the group is commutative. Like this:
(a * b)^2 = (a * b) * (a * b)
= (a * b) * (b * a) // Commutative property
= a * ((b * b) * a) // Associative property
= a * (a * (b * b)) // Commutative property
= (a * a) * (b * b) // Associative Property
Hence the given group is Abelian group. Hence Option C is the correct answer.