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enter image description here

Can somebody explain what this notation means?

Loobear23
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    https://www.google.com/url?sa=t&rct=j&q=&esrc=s&source=web&cd=11&cad=rja&uact=8&ved=2ahUKEwie3rqstrHoAhUCM6wKHSOdACsQFjAKegQIAxAB&url=https%3A%2F%2Fen.wikipedia.org%2Fwiki%2FAbelian_group&usg=AOvVaw3TfB-xf-lnD1X-K_m1J7T5, https://www.google.com/url?sa=t&rct=j&q=&esrc=s&source=web&cd=10&cad=rja&uact=8&ved=2ahUKEwie3rqstrHoAhUCM6wKHSOdACsQFjAJegQIAhAB&url=https%3A%2F%2Fen.wikipedia.org%2Fwiki%2FPresentation_of_a_group&usg=AOvVaw2IHcxX9nix2C6L6HN-_wlu – amWhy Mar 23 '20 at 20:23
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    https://math.stackexchange.com/questions/439014/presentation-of-abelian-group?rq=1 – amWhy Mar 23 '20 at 20:24
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    The first thing to understand is a presentation of a group in terms of generators and relations. This is slightly different for an abelian group $G$ because one conventionally uses additive rather than multiplicative notation (for a group known to be abelian). – hardmath Mar 23 '20 at 20:26
  • expressions to the right of | are equated to the identity element – J. W. Tanner Mar 23 '20 at 20:27
  • @amWhy None of those links address this question. Abelian groups have a free object, and we get presentations this way. Linking to Wikipedia articles for "abelian group" and "presentation of a group" don't cover this. – user1729 Mar 23 '20 at 20:58
  • @amWhy Yes, and the notation is of a presentation of an abelian group, not of a group. The group presentation would have the additional relators $[x_1, x_2]$, etc. – user1729 Mar 23 '20 at 21:01
  • @amWhy Having skimmed the article and searched for the word "presentation" I cannot find this section. – user1729 Mar 23 '20 at 21:05
  • @amWhy the nearest thing I can find is the second paragraph of the "finitely generated" section, which is really just the definition of a free object. – user1729 Mar 23 '20 at 21:09
  • I have voted to reopen this question as the offered links do not address the question, and because I can see no other way of the OP getting help...! – user1729 Mar 23 '20 at 21:13
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    @user1729: I'm in agreement that the offered links do not address the Question enough to serve as an Answer, but the Question itself is poorly presented for a Community Member with more than 400 reputation. The post consists of a fragmentary image and a summary adjuration for someone to explain it. I don't believe the OP has addressed the Comment I made earlier, but I'd join you in voting to reopen if an edit clarified which definition(s) the user needs to be articulated. – hardmath Mar 23 '20 at 21:22
  • @hardmath Well, your comment has the same issue as amWhy's links - the subtlety is that we are considering this as a quotient of a free abelian group rather than of a free group (which is not clear from your comment). I'm also unsure what extra clarity is required (although maybe my criticism here is with the new closure reasons). Could you maybe add a few sentences saying what you hope the OP could do to improve their question? – user1729 Mar 23 '20 at 21:33
  • The links, @user1729, are comments. Not an answer. And the closure of a question does not require that any answer in an answer field, nor an answer in a comment completely answer an ill-posed question evidencing absolutely no thought, research, or effort. Please understand this, because your comments are misleading in that you are essentially stating that only answered questions, no matter how poorly presented, can't be closed. That is simply not true. – amWhy Mar 23 '20 at 21:49
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    Which details should I add to the question so that it’s not lacking detail? – Loobear23 Mar 23 '20 at 21:54
  • How to ask a good question. – amWhy Mar 23 '20 at 21:57
  • @Loobear23: You've been around a while, so I'd hope you can take a look at the Question from the perspective of your Readers. Do you know what a presentation of a group is? If so, do you know how to relate the notation (shown in the part (b) of the image you posted) to generators and relations? I assume your difficulty lies somewhere in between these points, and it will expedite a useful response if you pin it down. – hardmath Mar 23 '20 at 22:25
  • @amWhy My point was that voting to close a question and simultaneously posting three links implies that the answer to it is easily found, which is not the case. Trying to answer this question by oneself is not easy; it requires some sophistication and knowledge of free objects, and goes beyond just "group presentations". – user1729 Mar 25 '20 at 12:58

2 Answers2

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Every finitely generated abelian group $A$ is the quotient of some free abelian group of the form $\mathbb Z^{\oplus n}$. If the kernel of this quotient map is again finitely generated as abelian group by elements $r_1,...,r_k$, which are all of the form $r = \sum \limits_{i=1}^n \alpha_i e_i$ with $\alpha_i \in \mathbb Z$, we can (analogously to group presentations) present $A$ as $\langle e_1,...,e_n \mid r_1,...,r_k\rangle$.

Jonas Linssen
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You have to take the free Abelian group $F$ on generator set $\{x_1,x_2,x_3\}$ (which is isomorphic to $\Bbb Z^3$) and take its quotient by the subgroup $H$ generated by the given elements, so in particular we will have $$4[x_1]+4[x_2]+8[x_3]\ =\ 0$$ in this group $G=F/H$, and the other two elements give rise to similar equations.

Berci
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