Translate of a subset of a group ($gA,Ag$)

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.