Left cosets are all in bijection


by Patrick Stevens Jun 17 2016 updated Jun 17 2016

The left cosets of a subgroup in a parent group are all the same size.

Let $~$H$~$ be a subgroup of $~$G$~$. Then for any two left cosets of $~$H$~$ in $~$G$~$, there is a Bijective function between the two cosets.


Let $~$aH, bH$~$ be two cosets. Define the function $~$f: aH \to bH$~$ by $~$x \mapsto b a^{-1} x$~$.

This has the correct codomain: if $~$x \in aH$~$ (so $~$x = ah$~$, say), then $~$ba^{-1} a x = bx$~$ so $~$f(x) \in bH$~$.

The function is injective: if $~$b a^{-1} x = b a^{-1} y$~$ then (pre-multiplying both sides by $~$a b^{-1}$~$) we obtain $~$x = y$~$.

The function is surjective: given $~$b h \in b H $~$, we want to find $~$x \in aH$~$ such that $~$f(x) = bh$~$. Let $~$x = a h$~$ to obtain $~$f(x) = b a^{-1} a h = b h$~$, as required.