Some of the things I've learned every day since Oct 10, 2016
104: Bijectivity of Left- or Right-Multiplication by Group Elements
January 30, 2017Posted by on
Let be a group and be any element of that group. Then the functions
Consider the function . By the left-cancellation property of groups, if , then . This means that if , then , and it follows that is injective. Since the domain and codomain of are the same (), this means that is in fact bijective.
The proof for bijectivity of is identical, but uses the right-cancellation property instead.
The intuitive significance of this is that left- or right-multiplication by a given element always sends distinct elements to distinct elements, which is pretty neat.