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Let $G$ be any abelian group, and let $\triangle_G = \{(g,g)\mid g\in G\}\subset G\times G.$

Is there any significance in studying the quotient group ${(G\times G)}\left/{\triangle_G}\right.?$ If so, where does the study such an object naturally occur?

Motivation for such a question: Let $F$ be a field not of characteristic 2. Let $F^\ast=F\setminus\{0\},$ be the multiplicative group of units of $F.$ Define $G=F^\ast/(F^\ast)^2.$

The motivation for such a question comes from the study of maps $q:G\times G \longrightarrow Br(F),$ where $Br(F)$ denotes the Brauer group of $F.$ If $-1$ is a square in $F,$ that is if $-1\in (F^\ast)^2,$ then $q(x,x)=0\in Br(F)$ for all $x\in G.$ Since $q(x,x)=0,$ this motivates why I'd like to mod out by $\triangle_G.$

Example: In the event that $F=\Bbb{Q}_2[\sqrt{-1}]$ we have that $Br(F)\cong \Bbb{Z}/2\Bbb{Z},$ and we can view $G$ as a 4-dimensional vector space over $\Bbb{Z}/2\Bbb{Z}.$ One such map $q:G\times G \longrightarrow \Bbb{Z}/2\Bbb{Z}$ is the symplectic bilinear form. Another is the constant map $q := 0\in Br(F),$ although this one is not very interesting.

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The group $H={(G\times G)}\left/{\triangle_G}\right.$ is just canonically isomorphic to $G$ itself. Indeed, the homomorphism $f:G\times G\to G$ defined by $f(g,h)=g-h$ is surjective and has $\triangle_G$ as its kernel, and thus induces an isomorphism from $H$ to $G$.

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  • $\begingroup$ Would you happen to know under exactly what circumstance one might study this object, though? The map which carries $(g,h)\mapsto g-h$ would more generally be the assignment $(g,h)\mapsto gh^{-1}.$ But where might $g-h$ come up naturally? $\endgroup$ Commented May 14, 2018 at 22:13
  • $\begingroup$ Um, any time you want to study subtraction on a group? That arises naturally whenever you want to study whether two group elements are equal, by subtracting them and seeing if you get $0$. $\endgroup$ Commented May 14, 2018 at 23:49
  • $\begingroup$ Could you be more specific? Is there something in terms of a trace map, possibly? I'm looking for context, rather than simply measuring when elements are equal. $\endgroup$ Commented May 15, 2018 at 18:40
  • $\begingroup$ I think you're vastly overthinking this. The map $(g,h)\mapsto g-h$ literally comes up ALL the time, whenever you compare two group elements. People just don't usually talk about the quotient $(G\times G)/\triangle_G$ because they immediately identify it with $G$ itself. $\endgroup$ Commented May 15, 2018 at 18:46
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    $\begingroup$ @Bumblebee: If $G$ is nonabelian then $\triangle_G$ is not normal in $G\times G$ so the quotient does not make sense. $\endgroup$ Commented Jul 4, 2022 at 12:50

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