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I'm having trouble finding when group multiplications are isomorphs beetween them or not.

Ex 1: Give examples of $3$ subgroups of $\mathbb{Z}_{27}\times\mathbb{Z}_{9}\times\mathbb{Z}_{3}$ that have order $27$ and are NOT isomorphs beetween them.

With order $27$ there are these subgroups:

  • $\mathbb{Z}_{27}\times\{0_9\}\times\{0_3\}$
  • $\{0_{27}\}\times\mathbb{Z}_{9}\times\mathbb{Z}_{3}$
  • $3\mathbb{Z}_{27}\times\{0_9\}\times\mathbb{Z}_{3}$
  • $9\mathbb{Z}_{27}\times3\mathbb{Z}_{9}\times\mathbb{Z}_{3}$

Aren't they all already NOT isomorphs beetween them??

Ex 2: Give examples of $4$ abelian groups with order $108$ that are NOT isomorphs beetween them

$108=2^2\times3^3$

So there these groups:

  • $\mathbb{Z}_{2}\times\mathbb{Z}_{2}\times\mathbb{Z}_{3}\times\mathbb{Z}_{3}\times\mathbb{Z}_{3}$
  • $\mathbb{Z}_{2}\times\mathbb{Z}_{2}\times\mathbb{Z}_{9}\times\mathbb{Z}_{3}$
  • $\mathbb{Z}_{2}\times\mathbb{Z}_{2}\times\mathbb{Z}_{27}$
  • $\mathbb{Z}_{4}\times\mathbb{Z}_{3}\times\mathbb{Z}_{3}\times\mathbb{Z}_{3}$
  • $\mathbb{Z}_{4}\times\mathbb{Z}_{9}\times\mathbb{Z}_{3}$
  • $\mathbb{Z}_{4}\times\mathbb{Z}_{27}$

Aren't they all Not isomorphs beetween them?

Thanks in advance.

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    Of your order $27$ subgroups, number 2) and 3) are isomorphic. However, 1) is the only one with any single elements of order $27$, and only 4) has _no_ elements of order $9$, so you do have $3$ non-isomorphics there. – 2017-01-30
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    How do know that 2) and 3) are isomorphic? Thanks – 2017-01-30
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    Because $(0, m, n)\mapsto (3m, 0, n)$ is an isomorphism. You should check that, by the way. – 2017-01-30
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    @Arthur Thanks, I think I understand Ex 1 .. Btw, do you know if in Ex:2 , are all they not isomorphic beetween them? – 2017-01-30
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    Those are all non-isomorphic, yes. Haven't you been told the classification of finitely generated abelian groups? It tells you _exactly_ what you need in order to answer these things for yourself, without any problems. – 2017-01-30

1 Answers 1

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There is only one simple rule to take into account namely :$$\Bbb Z_{mn} \cong \Bbb Z_m \times \Bbb Z_n \iff \gcd(m,n) = 1 $$

There is a simple theorem that says that ever abelian group can be written in the form $\Bbb Z_{n_1} \times \Bbb Z_{n_2} \times \ldots \times \Bbb Z_{n_k} $ where $n_1 \mid n_2 \mid \ldots \mid n_k $. In your examples this gives

$\begin{array} \Bbb Z_3 \times \Bbb Z_6 \times \Bbb Z_6\\ \Bbb Z_6 \times \Bbb Z_{18}\\ \Bbb Z_2 \times \Bbb Z_{54}\\ \Bbb Z_3 \times \Bbb Z_3 \times \Bbb Z_{12}\\ \Bbb Z_3 \times \Bbb Z_{36}\\ \Bbb Z_{108}\end{array}$

Note that this way of presenting is unique.