Question

Let G be a group and H ≤ G . Prove that the only right coset of H in G that is a
subgroup of G is H itself.

Answer 1

Answer on Question #81004 – Math – Abstract Algebra

Question

Let $G$ be a group and $\leq G$. Prove that the only right coset of $H$ in $G$ that is a subgroup of $G$ is $H$ itself.

Solution

By definition, $Hg = \{hg: h \in H\}$ is the right coset of $H$ in $G$ with respect to $g \in G$. Assume that $Hg$ is a subgroup.

According to the definition of a group, $1 \in Hg$. Therefore, $g^{-1} = 1 * g^{-1} = h_1 gg^{-1} = h_1 \in H$. According to the definition of a group, $g = (g^{-1})^{-1} \in H$, and for this reason $Hg = H$ (in this case, $hg \in H$ and $h = (hg^{-1})g \in Hg$, i.e. $Hg \subset H$ and $H \subset Hg$).

Answer

If $Hg$ is a subgroup, then $g \in H$ and $Hg = H$.

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