Explanation:

A Prime order group is a group in which the order of the group is a prime number. In this case, we need to determine whether a Prime order group has proper subgroups or not.

Definition:
A proper subgroup of a group G is a subgroup that is not equal to G itself.

Proof:
To show that a Prime order group has no proper subgroups, we need to consider the definition of a subgroup and the properties of a Prime order group.

1. Definition of a subgroup:
A subgroup H of a group G is a non-empty subset of G that is closed under the group operation and the inverse operation. In other words, if a and b are elements of H, then the product ab and the inverse a^(-1) are also elements of H.

2. Properties of a Prime order group:
- A Prime order group G has only two subgroups: the trivial subgroup {e} (containing only the identity element) and G itself.
- The order of the subgroup H of a group G must divide the order of G. In other words, if the order of G is p (a prime number), then the order of any subgroup H of G must be 1 (trivial subgroup) or p (G itself).

Proof by contradiction:
Assume that there exists a proper subgroup H of a Prime order group G. Since H is a subgroup of G, the order of H must divide the order of G (which is a prime number). This means that the order of H can only be 1 or p.

- If the order of H is 1, then H is the trivial subgroup {e}.
- If the order of H is p, then H is equal to G itself.

In either case, H is not a proper subgroup, which contradicts our assumption. Therefore, a Prime order group has no proper subgroups.

Conclusion:
The correct answer is option A: No Proper Subgroup. A Prime order group does not have any proper subgroups.