Question

The number of non-isomorphic abelian groups of order $2^2 \cdot 3^3 \cdot 5^4$ is __________.

Hint:

To determine the number of non-isomorphic abelian groups for a given order, use the partition theory for each prime factor in the prime factorization of the order.

Answer 1

Correct Answer: 30

The number of non-isomorphic abelian groups of a given order can be determined by considering the prime factorization of the order.
Given the order is $2^2 \cdot 3^3 \cdot 5^4$, we will separately consider the number of non-isomorphic abelian groups for each prime factor.
1. For $2^2$, the number of non-isomorphic abelian groups is determined by the partitions of 2. The partitions of 2 are: \[ 2 = 2 \quad \text{or} \quad 2 = 1 + 1 \] Thus, there are 2 non-isomorphic abelian groups for $2^2$: \[ \mathbb{Z}_4, \mathbb{Z}_2 \times \mathbb{Z}_2 \] 2. For $3^3$, the number of non-isomorphic abelian groups is determined by the partitions of 3. The partitions of 3 are: \[ 3 = 3 \quad \text{or} \quad 3 = 2 + 1 \quad \text{or} \quad 3 = 1 + 1 + 1 \] Thus, there are 3 non-isomorphic abelian groups for $3^3$: \[ \mathbb{Z}_27, \mathbb{Z}_9 \times \mathbb{Z}_3, \mathbb{Z}_3 \times \mathbb{Z}_3 \times \mathbb{Z}_3 \] 3. For $5^4$, the number of non-isomorphic abelian groups is determined by the partitions of 4. The partitions of 4 are: \[ 4 = 4 \quad \text{or} \quad 4 = 3 + 1 \quad \text{or} \quad 4 = 2 + 2 \quad \text{or} \quad 4 = 2 + 1 + 1 \quad \text{or} \quad 4 = 1 + 1 + 1 + 1 \] Thus, there are 5 non-isomorphic abelian groups for $5^4$: \[ \mathbb{Z}_{625}, \mathbb{Z}_{125} \times \mathbb{Z}_5, \mathbb{Z}_{25} \times \mathbb{Z}_{25}, \mathbb{Z}_{25} \times \mathbb{Z}_5 \times \mathbb{Z}_5, \mathbb{Z}_5 \times \mathbb{Z}_5 \times \mathbb{Z}_5 \times \mathbb{Z}_5 \] Now, the total number of non-isomorphic abelian groups of order $2^2 \cdot 3^3 \cdot 5^4$ is the product of the individual counts for each prime factor: \[ 2 \times 3 \times 5 = 30 \] Thus, the number of non-isomorphic abelian groups of order $2^2 \cdot 3^3 \cdot 5^4$ is \(\boxed{30}\).