Question #146436
Find all Abelian groups of cardinality 36, and all Abelian groups of cardinality 54.
1
Expert's answer
2020-11-26T19:38:51-0500

For the abelian groups of order 36 :

36=223236=2^2 3^2

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 :

  1. There are elements of order 4 and 9. In this case their product gives un an element of order 36 and thus GZ/36ZG\simeq \mathbb{Z}/36\mathbb{Z}
  2. There is an element of order 4, but no element of order 9. Therefore there is an element of order 18 (say gg) and an element of order 2 which is not in <g><g> (due to cardinality reasons). Therefore we can decompose GG in a direct product : GZ/18Z×Z/2ZG\simeq \mathbb{Z}/18\mathbb{Z} \times \mathbb{Z}/2\mathbb{Z} .
  3. There is an element of order 9, but no element of order 4. This case is completely symmetrical and we have a product GZ/12Z×Z/3ZG \simeq \mathbb{Z}/12\mathbb{Z}\times\mathbb{Z}/3\mathbb{Z} .
  4. 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 gg , and there is another element of order 6 which is not in <g><g> Therefore our product is G(Z/6Z)2G\simeq (\mathbb{Z}/6\mathbb{Z})^2

For abelian groups of order 54=23354=2\cdot3^3 we have essentially the same algorithm:

There are forcefully the elements of order 2 and 3. Now there are cases:

  1. 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 GZ/54ZG\simeq\mathbb{Z}/54\mathbb{Z} .
  2. 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 gg and there is an element of order 3 that is not in <g><g> . Thus we can decompose GZ/2Z×(Z/9Z×Z/3Z)Z/18Z×Z/3ZG\simeq \mathbb{Z}/2\mathbb{Z}\times (\mathbb{Z}/9\mathbb{Z}\times\mathbb{Z}/3\mathbb{Z}) \simeq \mathbb{Z}/18\mathbb{Z} \times \mathbb{Z}/3\mathbb{Z} .
  3. There are no elements of order 9. In this case there is an element of order 6, we decompose GG into a product GZ/6Z×HG\simeq \mathbb{Z}/6\mathbb{Z} \times H for HH a group of cardinal 9 without any element of order 9, so this is (Z/3Z)2(\mathbb{Z}/3\mathbb{Z})^2 . Therefore we have GZ/6Z×(Z/3Z)2G\simeq \mathbb{Z}/6\mathbb{Z}\times (\mathbb{Z}/3\mathbb{Z})^2 .

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