Group theory/cosets: Prove or disprove the map $\phi: L_{H}\to R_{H}$ defined by $\phi(gH)=Hg$ is a function.

Question

Note that $H$ is a subgroup of a group $G$ and $L_{H}$ is the left cosets of $H$ in $G$ and $R_{H}$ is the right cosets of $H$ in $G$.

How do I show that this map is well-defined or not?

What I tried:

If $g_{1}H=g_{2}H$, then $\phi(g_{1}H)=Hg_{1}=Hg_{2}=\phi (g_{2}H)$, but I don't think this is true in general.

Answer 1

Your question is equivalent to the statement that "$gH=hH\Rightarrow Hg=Hh$". This does not hold in general. To see this, recall that $gH=hH$ if and only if $g^{-1}h\in H$, while $Hg=Hh$ if and only if $gh^{-1}\in H$. Then it is not necessarily the case that "$gh^{-1}\in H\Leftrightarrow g^{-1}h\in H$".

For example, take $H=\langle (12)\rangle\leq S_3$. Then $(123)H=(23)H$ while $H(123)\neq H(23)$. (Instead, $H(23)=H(132)$.)

The map you actually want is $gH\mapsto Hg^{-1}$. Then this map is well defined.