Proceed as in Example 11.13 to find all abelian groups, up to isomorphism, of the given order.

Order 720

Data from example 11.13

Find all abelian groups, up to isomorphism, of order 360. The phrase up to isomorphism signifies that any abelian group of order 360 should be structurally identical (isomorphic) to one of the groups of order 360 exhibited.

We make use of Theorem 11.12. Since our groups are to be of the finite order 360, no factors Z will appear in the direct product shown in the statement of the theorem. First we express 360 as a product of prime powers 2³ 3² 5. Then using Theorem 11.12, we get as possibilities

1. Z₂ x Z₂ x Z₂ x Z₃ x Z₃ x Z₅ 

2. Z₂ x Z₄ x Z₃ x Z₃ x Z₅ 

3. Z₂ x Z₂ x Z₂ x Z₉ x Z₅ 

4. Z₂ x Z4 x Z9 x Z₅ 

S. Z₅ x Z3 x Z3 x Z₅

6. Z₅ x Z₉ x Z₅

Thus there are six different abelian groups (up to isomorphism) of order 360.