\begin{proof} Suppose that the right coset $Hg$ equals some left coset $k H$, where $k \in G$, so that $Hg = kH$. Then $g \in kH$, thus $gH = kH$. This shows that $Hg = gH$, or equivalently $gHg^{-1} = H$, so $g \in N_G(H)$. \end{proof}

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Abstract Algebra

Exercise 3.2.6 (If $gH = Hk$ for some $k \in G$, then $g \in N_G(H)$)

Proof. Suppose that the right coset $𝐻𝑔$ equals some left coset $𝑘𝐻$ , where $k \in G$ , so that $𝐻𝑔 = 𝑘𝐻$ . Then $g \in 𝑘𝐻$ , thus $𝑔𝐻 = 𝑘𝐻$ . This shows that $𝐻𝑔 = 𝑔𝐻$ , or equivalently $𝑔𝐻 g^{- 1} = H$ , so $g \in N_{G} (H)$ . □

Exercise 3.2.6 (If $gH = Hk$ for some $k \in G$, then $g \in N_G(H)$)

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