# Today I Learned

Some of the things I've learned every day since Oct 10, 2016

## 104: Bijectivity of Left- or Right-Multiplication by Group Elements

Let $G$ be a group and $x$ be any element of that group. Then the functions

$f: a \mapsto xa$

$g: a \mapsto ax$

are bijective.

#### Proof:

Consider the function $f$. By the left-cancellation property of groups, if $ab = ac$, then $b = c$. This means that if $b \neq c$, then $ab \neq ac$, and it follows that $f$ is injective. Since the domain and codomain of $f$ are the same ($G$), this means that $f$ is in fact bijective.

The proof for bijectivity of $g$ 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.