Let $H$ be a subgroup of $G$. If for each $a \in G$ there exists $b \in G$ such that $aH = Hb$, show that $H$ is a normal subgroup of $G$.

Im tutoring a person in first year abstract algebra, Im having a trouble getting this to work, I know theres some little detail or catch im missing. I can get to the point where all we need to show is $ba^{-1} \in H$ to prove $aHa^{-1} = H$ (abusive notation).


For right cosets we have

$Hx=Hy \iff x\in Hy \iff xy^{-1}\in H \iff yx^{-1}\in H \iff y\in Hx$.

From the given assumption, if $aH=Hb$, then we have $a\in aH=Hb$, so that, by the above, $Hb=Ha$, and that means

$aH=Ha\ \iff\ aHa^{-1}=H$.


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