= Translate of a subset of a group
{title2=$gA,\quad Ag$}
= Translate
{synonym}
For a <subset> $A$ of a <group> $G$ and $g\in G$, its left translate is $gA=\{ga:a\in A\}$ and its right translate is $Ag=\{ag:a\in A\}$. In an additive <abelian group>, these are the same translated <set> $g+A$. Multiplication by $g$ is a <bijection>, so translation preserves <cardinality>.
Left and right translates can differ in a noncommutative <group>. In the symmetric group $S_3$, take $A=\{1,(12)\}$ and $g=(123)$, composing permutations right to left. Then $gA=\{(123),(13)\}$ while $Ag=\{(123),(23)\}$. When $A$ is a <subgroup>, these translated <sets> are precisely left and right <cosets>. Translation lets <additive combinatorics> compare configurations without treating their absolute location as part of their shape.
Back to article page
