Editing Group (mathematics) (section)

=== <span id="translation"></span> Division ===
Given elements <math>a</math> and <math>b</math> of a group {{tmath|1= G }}, there is a unique solution <math>x</math> in <math>G</math> to the equation {{tmath|1= a\cdot x=b }}, namely {{tmath|1= a^{-1}\cdot b }}.{{efn|One usually avoids using fraction notation <!--use {{math}}, since <math> in footnotes is unreadable on mobile devices-->{{math|{{sfrac|''b''|''a''}}}} unless {{math|''G''}} is abelian, because of the ambiguity of whether it means {{math|''a''<sup>−1</sup> ⋅ ''b''}} or {{math|''b'' ⋅ ''a''<sup>−1</sup>}}.)}}{{sfn|Artin|2018|p=40}} It follows that for each <math>a</math> in {{tmath|1= G }}, the function <math>G\to G</math> that maps each <math>x</math> to <math>a\cdot x</math> is a [[bijection]]; it is called ''left multiplication'' by <math>a</math> or ''left translation'' by {{tmath|1= a }}.

Similarly, given <math>a</math> and {{tmath|1= b }}, the unique solution to <math>x\cdot a=b</math> is {{tmath|1= b\cdot a^{-1} }}. For each {{tmath|1= a }}, the function <math>G\to G</math> that maps each <math>x</math> to <math>x\cdot a</math> is a bijection called ''right multiplication'' by <math>a</math> or ''right translation'' by {{tmath|1= a }}.