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I'm a bit confused between the relationship between cross and direct sum. For example if I wanted to put $T =\mathbb Z_2 \times \mathbb Z_2 \times \mathbb Z_3 \times \mathbb Z_4 \times \mathbb Z_9 \times \mathbb Z_9$ into the form $T \cong \mathbb Z_n \oplus ... \oplus \mathbb Z_m $ how would I go about doing this?

I believe that $\mathbb Z_2 \times \mathbb Z_9 \cong \mathbb Z_{18}$ and $\mathbb Z_3 \times \mathbb Z_4 \cong \mathbb Z_{12}$ so can I just say that $T \cong \mathbb Z_{12} \oplus \mathbb Z_{18} \oplus \mathbb Z_{18} $?

user2973447
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  • You are confusing two different things. The direct sum is like the direct product except that only finitely many coordinates of an element can be nonzero. The question you're asking is how to roll up the subscripts of a finitely generated Abelian group. https://en.wikipedia.org/wiki/Finitely_generated_abelian_group – user4894 Mar 09 '17 at 20:33

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For a finite number of factors, the direct sum $\oplus$ and the direct product $\times$ of abelian groups is isomorphic - see here. So we have $$ T =\mathbb Z_2 \times \mathbb Z_2 \times \mathbb Z_3 \times \mathbb Z_4 \times \mathbb Z_9 \times \mathbb Z_9\cong \mathbb Z_{12} \times \mathbb Z_{18} \times \mathbb Z_{18} \cong \mathbb Z_{12} \oplus \mathbb Z_{18} \oplus \mathbb Z_{18}. $$

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Dietrich Burde
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