Statement a: Every cyclic group is abelian.

A group is said to be cyclic if it is generated by a single element. In other words, there exists an element in the group such that every other element of the group can be obtained by repeatedly applying the group operation to this element.

To prove that every cyclic group is abelian, let's consider a group G generated by a single element a. For any two elements x and y in G, we have to show that xy = yx.

Since G is cyclic, we can express any element x in G as a^n, where n is an integer. Similarly, y can be expressed as a^m.

Now, let's consider the product xy:
xy = (a^n)(a^m) = a^(n+m)

Similarly, let's consider the product yx:
yx = (a^m)(a^n) = a^(m+n)

Since addition of integers is commutative, we have n+m = m+n. Therefore, a^(n+m) = a^(m+n).

Hence, xy = yx, and we have shown that every cyclic group is abelian.

Statement b: Every group of prime order is cyclic.

A group of prime order is a group that has a number of elements equal to a prime number. To prove that every group of prime order is cyclic, we can use Lagrange's theorem.

Lagrange's theorem states that the order of any subgroup of a group divides the order of the group. Therefore, the order of any element in a group divides the order of the group.

Now, let's consider a group G of prime order p. Since p is prime, the only positive divisors of p are 1 and p. Therefore, the order of any element in G must be either 1 or p.

If there exists an element a in G such that the order of a is p, then G is cyclic since it is generated by a single element.

If all elements in G have an order of 1, then G consists of only the identity element. In this case, G is trivially cyclic since it is generated by the identity element.

Therefore, every group of prime order is cyclic.

Statement c: Every abelian group is cyclic.

This statement is false. There exist abelian groups that are not cyclic. One example is the direct product of two cyclic groups of different orders. For example, the direct product of Z2 and Z3 is abelian but not cyclic.

Therefore, option c is false.

Statement d: None of these

Option d is incorrect because statements a and b are true.