# Left cosets are all in bijection

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.

# Proof

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.

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