Abelian Groups of Order 4

To determine the number of different non-isomorphic Abelian groups of order 4, we need to consider the possible structures of such groups.

Abelian Groups
An Abelian group, also known as a commutative group, is a group in which the group operation is commutative. In other words, for any two elements a and b in the group, a * b = b * a.

Order of a Group
The order of a group is the number of elements in the group. For this question, we are considering Abelian groups of order 4.

Different Non-Isomorphic Groups
To determine the different non-isomorphic Abelian groups of order 4, we need to consider the possible structures of such groups.

Prime Factorization of 4
The prime factorization of 4 is 2 * 2.

Direct Product of Groups
The direct product of two groups, G and H, denoted as G x H, is a new group formed by taking all possible pairs of elements, one from G and one from H, and defining a new group operation. This operation is defined component-wise, meaning that the operation between two pairs is done separately for each component.

Abelian Groups of Order 2
There is only one Abelian group of order 2, which is the cyclic group of order 2: {0, 1} under addition modulo 2.

Abelian Groups of Order 2 * 2 = 4
To determine the different non-isomorphic Abelian groups of order 4, we need to consider the possible combinations of two Abelian groups of order 2.

Direct Product of Cyclic Groups of Order 2
One possible combination is the direct product of two cyclic groups of order 2: C2 x C2. This group has 4 elements: {(0, 0), (0, 1), (1, 0), (1, 1)}. The group operation is defined component-wise.

Direct Product of Cyclic Group of Order 2 and Cyclic Group of Order 2
Another possible combination is the direct product of a cyclic group of order 2 and another cyclic group of order 2: C2 x C2. This group also has 4 elements: {0, 1, 2, 3}. The group operation is defined modulo 4.

Isomorphism of Groups
Two groups are isomorphic if there is a one-to-one correspondence between their elements that preserves the group operation. In other words, two groups are isomorphic if they have the same structure.

Number of Different Non-Isomorphic Groups
In this case, the two combinations mentioned above are isomorphic to each other. Therefore, there are only 2 different non-isomorphic Abelian groups of order 4.

Correct Answer
The correct answer is option 'A' - 2.