For the abelian groups of order 36 :
36=2232
We will try to decompose this group into a product and using the Chinese remainder theorem to regroup all the possible decompositions.
So there are forcefully elements of order 2 and 3 respectively. Now there are different cases :
- There are elements of order 4 and 9. In this case their product gives un an element of order 36 and thus G≃Z/36Z
- There is an element of order 4, but no element of order 9. Therefore there is an element of order 18 (say g) and an element of order 2 which is not in <g> (due to cardinality reasons). Therefore we can decompose G in a direct product : G≃Z/18Z×Z/2Z .
- There is an element of order 9, but no element of order 4. This case is completely symmetrical and we have a product G≃Z/12Z×Z/3Z .
- If there is no element of order 4 or 9, we still have an element of order 6 (by multiplying the elements of order 2 and 3), say g , and there is another element of order 6 which is not in <g> Therefore our product is G≃(Z/6Z)2
For abelian groups of order 54=2⋅33 we have essentially the same algorithm:
There are forcefully the elements of order 2 and 3. Now there are cases:
- There is an element of order 27. In this case the product of elements of order 2 and 27 will give an element of order 54 and thus G≃Z/54Z .
- There is no element of order 27, but there is an element of order 9. In this case there is an element of order 18, say g and there is an element of order 3 that is not in <g> . Thus we can decompose G≃Z/2Z×(Z/9Z×Z/3Z)≃Z/18Z×Z/3Z .
- There are no elements of order 9. In this case there is an element of order 6, we decompose G into a product G≃Z/6Z×H for H a group of cardinal 9 without any element of order 9, so this is (Z/3Z)2 . Therefore we have G≃Z/6Z×(Z/3Z)2 .
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