3
$\begingroup$

Let $A$ be a abelian group generated by elements $\langle a_1,a_2,a_3\rangle$ and $B$ be a subgroup generated by $\langle b_1,b_2,b_3\rangle$ where $\begin{pmatrix} b_1\\ b_2\\b_3 \end{pmatrix}= \begin{pmatrix} a_1&a_2&0\\ a_1&0&a_3\\ 0&a_2&a_3 \end{pmatrix}\begin{pmatrix}\alpha\\ \beta\\ \gamma\end{pmatrix}$ and $\alpha,\beta, \gamma \in \mathbb Z$

How then might we express $A/B$ as $\bigoplus_i \mathbb Z_{m_i}$?

  • 1
    @Arturo, I hope you find that$5$that you lost in your head - you might need it some day.2012-03-19

1 Answers 1

1

You should find the Smith decomposition of the given matrix, which gives you exactly the needed order for the sequence of cyclic groups. A reference for this process: Algebra, by Michael Artin.

  • 0
    Please ignore the comment above, I can't seem to delete it.2012-03-17