To determine the order of the group, we need to find the number of elements in the set.

Step 1: Counting the number of elements in the set
The given set is {1, 2, 3, 4}. Counting the number of elements, we find that there are 4 elements in the set.

Step 2: Checking if the set forms a group under multiplication modulo n
To determine if the set forms a group under multiplication modulo n, we need to check the following conditions:
1. Closure: For any two elements a and b in the set, a * b (mod n) should also be in the set.
2. Associativity: For any three elements a, b, and c in the set, (a * b) * c (mod n) should be equal to a * (b * c) (mod n).
3. Identity: There should exist an identity element e in the set such that for any element a in the set, a * e (mod n) = a.
4. Inverse: For any element a in the set, there should exist an inverse element b in the set such that a * b (mod n) = e.

Checking the closure property:
1 * 1 (mod 4) = 1
1 * 2 (mod 4) = 2
1 * 3 (mod 4) = 3
1 * 4 (mod 4) = 0
2 * 1 (mod 4) = 2
2 * 2 (mod 4) = 0
2 * 3 (mod 4) = 2
2 * 4 (mod 4) = 0
3 * 1 (mod 4) = 3
3 * 2 (mod 4) = 2
3 * 3 (mod 4) = 1
3 * 4 (mod 4) = 0
4 * 1 (mod 4) = 0
4 * 2 (mod 4) = 0
4 * 3 (mod 4) = 0
4 * 4 (mod 4) = 0

From the above calculations, we can see that all the results are in the set {1, 2, 3, 4}. Therefore, the set satisfies the closure property.

Checking the associativity property:
Since multiplication is associative, the set satisfies the associativity property.

Checking the identity property:
There is no element e in the set {1, 2, 3, 4} such that a * e (mod 4) = a for all elements a in the set. Therefore, the set does not have an identity element and does not satisfy the identity property.

Checking the inverse property:
For each element in the set, we need to find an inverse element such that the product of the element and its inverse is congruent to the identity element modulo 4. However, since the set does not have an identity element, it does not have inverse elements either.

Since the set does not satisfy all the conditions to be a group, it cannot be considered as an abelian group under multiplication modulo 4. Therefore, the correct answer is option B) 4, 5.