Given elements a and b of a group ⁠ G ⁠, there is a unique solution x in G to the equation ⁠ a ⋅ ⋅ x = b ⁠, namely ⁠ a − − 1 ⋅ ⋅ b ⁠. It follows that for each a in G , the function G → → G that maps each x to a ⋅ ⋅ x is a bijection ; it is called left multiplication by a or left translation by ⁠ a ⁠.