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I'm looking for an introduction to free abelian groups and their presentation, specifically to understand notation such as:

$G=^{ab}\left\langle a,b,c \text{ } | a + 2b + 3c = −a − 2b − 7c = 0 \right\rangle$

and how starting from this presentation, you would be able to obtain an isomorphism between G and the direct product of cyclic groups

Thanks in advance

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  • $\begingroup$ That is not a standard notation. Those two equalities, are they supposed to mean three different relations? Also the $X^{ab}$ typically should be after the group definition. Why is it attached to the equality sign? Very untidy notation. Where did you get that from? $\endgroup$
    – freakish
    Commented Aug 30, 2023 at 13:08
  • $\begingroup$ They're supposed to denote two different relations; I got this from some notes a professor from my university wrote $\endgroup$
    – iki
    Commented Aug 30, 2023 at 13:11
  • $\begingroup$ Again regarding the equalities, it's probably a quick way to write something such as ⟨𝑎,𝑏,𝑐 |𝑎+2𝑏+3𝑐 =0, −𝑎−2𝑏−7𝑐=0⟩ $\endgroup$
    – iki
    Commented Aug 30, 2023 at 13:16
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    $\begingroup$ What you want is "Smith Normal Form". You probably want a book on linear algebra or optimization algorithms, which should have the instructions for computing the Smith Normal Form. You can read off the invariant factor decomposition from the Smith Normal form. Search for "Smith Normal Form" in this site for some examples and explanations. $\endgroup$
    – Arturo Magidin
    Commented Aug 30, 2023 at 16:19
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    $\begingroup$ @freakish Adding the equations gives $-4c=0$, not $c=0$. The group is $\mathbb{Z} \oplus \mathbb{Z}/(4\mathbb{Z})$. $\endgroup$
    – Derek Holt
    Commented Aug 30, 2023 at 19:30

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