Abelian Groups of Order 20

Introduction:
An abelian group is a group in which the group operation is commutative. In other words, for any elements a and b in the group, the order in which the operation is performed does not change the result.

Explanation:
To determine the number of abelian groups of order 20, we need to consider the prime factorization of 20.

Prime Factorization of 20:
20 = 2^2 * 5

Structure of Abelian Groups:
Since abelian groups are commutative, the structure of an abelian group of order 20 will depend on the structure of its cyclic subgroups.

Cyclic Subgroups:
- A cyclic group is a group generated by a single element.
- The number of cyclic subgroups of a group of order n is equal to the number of divisors of n.

Number of Cyclic Subgroups:
For a group of order 20, the number of cyclic subgroups can be calculated as follows:
- Number of cyclic subgroups of order 1: 1 (identity subgroup)
- Number of cyclic subgroups of order 2: 1 (generated by an element of order 2)
- Number of cyclic subgroups of order 4: 2 (generated by an element of order 4)
- Number of cyclic subgroups of order 5: 2 (generated by an element of order 5)
- Number of cyclic subgroups of order 10: 2 (generated by an element of order 10)
- Number of cyclic subgroups of order 20: 1 (the whole group)

Combining Cyclic Subgroups:
We can combine these cyclic subgroups to form abelian groups. Since abelian groups are commutative, the combination of cyclic subgroups will not change the structure of the group.

Number of Abelian Groups:
The number of abelian groups of order 20 can be determined by considering the combinations of cyclic subgroups:
- We can combine the cyclic subgroups of order 2, 4, 5, and 10 in different ways to form different abelian groups.
- The number of combinations can be calculated as follows:
Number of combinations = (Number of cyclic subgroups of order 2 + 1) * (Number of cyclic subgroups of order 4 + 1) * (Number of cyclic subgroups of order 5 + 1) * (Number of cyclic subgroups of order 10 + 1)
= (1 + 1) * (2 + 1) * (2 + 1) * (2 + 1)
= 2 * 3 * 3 * 3
= 54

Conclusion:
The number of abelian groups of order 20, up to isomorphism, is 54. Therefore, none of the given options (a), (c), (d) are correct. The correct answer is option (b), which states that there are 2 abelian groups of order 20 up to isomorphism.