According to page 158 of Dummit and Foote's Abstract Algebra (3rd edition):

Theorem. (Fundamental Theorem of Finitely Generated Abelian Groups) Let $G$ be a finitely generated abelian group. Then

(1) $G \cong \mathbb{Z}^r \times Z_{n_1} \times Z_{n_2} \times ... \times Z_{n_s}$, for some integers $r$, $n_1$, $n_2$, ... , $n_s$ satisfying the following conditions: (a) $r \ge 0$ and $n_j \ge 2$ for all $j$, and (b) $n_{i+1} \mid n_i$ for $1 \le i \le s-1$

(2) the expression in (1) is unique: if $G\cong \mathbb{Z}^t \times Z_{m_1} \times Z_{m_2} \times ... \times Z_{m_u}$, where $t$ and $m_1$, $m_2$, ... , $m_u$ satisfy (a) and (b) (i.e., $t \ge 0$, $m_j \ge 2$ for all $j$ and $m_{i+1} \mid m_i$ for $1 \le i \le u-1$), then $t = r$, $u = s$ and $m_i = n_i$ for all $i$.

Question: If an abelian group is expressed as finite generators and their defining relations (for example $G = \langle s,t,u,v \mid s^{4}t^{2}u^{10}v^{6} = s^{8}t^{4}u^{8}v^{10} = s^{6}t^{2}u^{9}v^{8} = e_G\rangle$), how to compute and express the group as a direct sum of cyclic groups and find the corresponding generators satisfying the relations? Could someone explain the computing procedure and show some worked examples? Or could someone point out which textbook taught such computing procedure with worked examples and at which pages/sections?

P.S.1 Although I can't find which textbook taught such computing procedure, Derek Holt gave a link to a PDF document teaching such computing procedure in his comment.

P.S.2. Could someone reply my 2nd comment of lhf's answer and explain how to find that 4 generators in $C_2 \times C_2 \times C_{\infty}$ satisfying the relations?

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    $\begingroup$ After a bit of searching, I found this link, which seems to explain everything with worked examples. $\endgroup$ – Derek Holt Nov 3 '15 at 20:02
  • $\begingroup$ If |a| = $\infty$ then <a> ~ Z, if |a| = m then <a> ~ $Z_m$. So that fundamental expression is the same thing. $\endgroup$ – fleablood Nov 4 '15 at 21:07

The tool for this is the Smith Normal Form, a kind of Gaussian elimination for PID.

For the example $G = \langle s,t,u,v \mid s^{4}t^{2}u^{10}v^{6} = s^{8}t^{4}u^{8}v^{10} = s^{6}t^{2}u^{9}v^{8} = e_G \rangle$ written additively, the matrix is $$ \left( \begin{array}{cccc} 4 & 2 & 10 & 6 \\ 8 & 4 & 8 & 10 \\ 6 & 2 & 9 & 8 \\ \end{array} \right) $$ whose Smith Normal Form is $$ \left( \begin{array}{cccc} 1 & 0 & 0 & 0 \\ 0 & 2 & 0 & 0 \\ 0 & 0 & 2 & 0 \\ \end{array} \right) $$ as computed by Mathematica using the code mentioned here.

This means that $G$ is generated by four elements $g_1, g_2, g_3, g_4$ such that $1g_1=0$, $2g_2=0$, $2g_3=0$, and no restrictions on $g_4$, and so $G \cong C_1 \times C_2 \times C_2 \times C_{\infty} \cong C_2 \times C_2 \times C_{\infty}$.

$g_1, g_2, g_3, g_4$ are obtained from $s,t,u,v$ by applying the matrices $P$ and $Q$ such that $PAQ$ is the diagonal matrix above.

All this is explained in several books. One is Finitely Generated Abelian Groups and Similarity of Matrices over a Field by Christopher Norman. A shorter account appears in Jacobson's Basic Algebra I.

  • $\begingroup$ Thank you!! (1) Could you please explain a little more (or give a sketch of proof) why this computing will yield the result corresponding to the correct direct sum of cyclic groups? (Or which book contains the proof and at which pages/sections?) (2) How to tell/compute which elements in $C_{\infty} \times C_2 \times C_2$ correspond to the generators $s,t,u,v$? $\endgroup$ – TCHuang Nov 5 '15 at 17:18
  • $\begingroup$ By (2) I mean since $G \cong C_2 \times C_2 \times C_{\infty}$ and $G$ is generated by 4 elements satisfying those 3 relations, there must also be 4 elements in $C_2 \times C_2 \times C_{\infty}$ generating $C_2 \times C_2 \times C_{\infty}$ and satisfying those 3 relations. Is there a method to find such 4 elements in $C_2 \times C_2 \times C_{\infty}$? Thanks! $\endgroup$ – TCHuang Nov 5 '15 at 20:13
  • $\begingroup$ When will you reply my 2nd comment and explain how to find that 4 generators in $C_2 \times C_2 \times C_{\infty}$ satisfying the relations? $\endgroup$ – TCHuang Nov 10 '15 at 18:53
  • $\begingroup$ @Tien-ChengHuang, using the $P$ and $Q$ matrices I've mentioned. You have to perform the algorithm for computing the Smith Normal Form to find these matrices. $\endgroup$ – lhf Nov 10 '15 at 19:34

If G has generators {$a_1.... a_r$} where $|a_i|= \infty$ and {$b_1...b_t$} where $|b_i| = m_i$ then G = $\prod_{i=1}^r<a_i> \times \prod_{j=1}^t<b_j>$. But each $\langle a_i \rangle \cong \mathbb Z$ so $\prod_{i=1}^r \cong \mathbb Z^r$ and each $\langle b_j \rangle \cong \mathbb Z_{m_j}$ so $G \cong \mathbb{Z}^r \times Z_{m_1} \times Z_{m_2} \times ... \times Z_{m_t}$

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    $\begingroup$ In general, the relations you're given are not necessarily of the form $b_i^{m_i} = 1$. There is some linear algebra involved in finding nice generators. For example, if you are given generators $a$ and $b$ with relation $a^4 = b^2$ (and of course $ab = ba$), then how are you supposed to guess that you should really take $a$ and $a^{-2}b$ as your generators (of infinite order and order $2$, respectively)? $\endgroup$ – Remy Nov 5 '15 at 4:53

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