GATE CS 2007 Q21 — Monoids and Groups

To find the number of non-isomorphic Abelian groups of order $4$, we can use the Fundamental Theorem of Finitely Generated Abelian Groups.

For any group of order $p^k$ (where $p$ is a prime number), the number of non-isomorphic Abelian groups is equal to the number of integer partitions of the exponent $k$, denoted as $p(k)$.

Here, the order of the group is $4 = 2^2$. Thus, $p = 2$ and $k = 2$.

The partitions of $2$ are:

1. $2$ — corresponding to the cyclic group $\mathbb{Z}_4$
2. $1 + 1$ — corresponding to the direct product $\mathbb{Z}_2 \times \mathbb{Z}_2$ (the Klein four-group)

These two groups are not isomorphic because:

• $\mathbb{Z}_4$ contains an element of order $4$ (it is cyclic).
• $\mathbb{Z}_2 \times \mathbb{Z}_2$ contains no element of order $4$; every non-identity element has order $2$.

Therefore, there are exactly $2$ non-isomorphic Abelian groups of order $4$.