# Determining torsion coefficients

Determine the torsion coefficients of $\mathbb{Z}_6 \times \mathbb{Z}_6 \times \mathbb{Z}_{10}$.

Now I know that if I rearrange $\mathbb{Z}_6 \times \mathbb{Z}_6 \times \mathbb{Z}_{10}$ to the form $\mathbb{Z}_{m1} \times \mathbb{Z}_{m2} \times . . . \times \mathbb{Z}_{mk} \times \mathbb{Z}^s$ for $s \in \mathbb{N}$, where $s$ is the rank of $G$, with $G \cong \mathbb{Z}_6 \times \mathbb{Z}_6 \times \mathbb{Z}_{10}$. Then the torsion coefficients are $m_1, m_2,. . . , m_k$ (by the classification theorem).

But how do I rearrange this?

• If your statement was true then you have $\mathbb{Z}_6 \times \mathbb{Z}_6 \times \mathbb{Z}_{10} = \mathbb{Z}_6 \times \mathbb{Z}_6 \times \mathbb{Z}_{10} \times \mathbb{Z}^0$, so the torsions coefficients would be $6$, $6$ and $10$, which seems pretty strange to me. Could you provide us with the exact definition of torsion coefficients that you are given? Jan 22 '16 at 5:36
• @JendrikStelzner I'm being given them via the classification theorem. For which I have as: Any finitely generated abelian group $G$ is isomorphic to a direct product of cyclic groups$\mathbb{Z}_{m1} \times \mathbb{Z}_{m2} \times . . . \times \mathbb{Z}_{mk} \times \mathbb{Z}^s$ for $s \in \mathbb{N}$ where $m_1|m_2, m_2|m_3, . . . , m_{k-1}|m_k$, $s =$ rank of $G$ and $m_1, m_2, . . . , m_k$ are the torsion coefficients of $G$. Jan 22 '16 at 5:41

If $$n$$ and $$m$$ are coprime then $$\mathbb{Z}_n \times \mathbb{Z}_m \cong \mathbb{Z}_{nm}$$. Therefore $$\mathbb{Z}_6 \times \mathbb{Z}_6 \times \mathbb{Z}_{10} \cong \mathbb{Z}_2 \times \mathbb{Z}_3 \times \mathbb{Z}_2 \times \mathbb{Z}_3 \times \mathbb{Z}_2 \times \mathbb{Z}_5 \cong \mathbb{Z}_2^3 \times \mathbb{Z}_3^2 \times \mathbb{Z}_5.$$ By rearranging the factors (or rather summands) we get $$\mathbb{Z}_2^3 \times \mathbb{Z}_3^2 \times \mathbb{Z}_5 \cong \mathbb{Z}_2 \times (\mathbb{Z}_2 \times \mathbb{Z}_3) \times (\mathbb{Z}_2 \times \mathbb{Z}_3 \times \mathbb{Z}_5) \cong \mathbb{Z}_2 \times \mathbb{Z}_6 \times \mathbb{Z}_{30}.$$ So using the definition you provided in the comments the torsion coefficients are $$2, 6, 30$$ (as this is an example of a finite groups the rank is $$0$$).