Hw8 Problem 13

Problem 13

Let $H$ be a subgroup of a group $G$ with the property that $g^{-1}hg \in H$ for all $g\in G$ and all $h\in H$. Show that every left coset $gH$ is equal to the right coset $Hg$.

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Solution

Let $gh \in gH$ where $g \in G$ and $h \in H$. Then $gh = ghe = ghg^{-1}g = [(g^{-1})^{-1} h g^{-1}]g \in Hg$.

So, $gH \subseteq Hg$.

Let $hg \in Hg$, where $g \in G$ and $h \in H$. Then $hg = ehg = gg^{-1}hg = g(g^{-1}hg) \in gH$.

So, $Hg \subseteq gH$.

Since $Hg$ and $gH$ are subsets of each other $Hg=gH$.