TITLE: How to prove inverse property of $P = \{hk \mid h \in H, k \in K\}$ is a subgroup of G?
QUESTION [0 upvotes]: Let $G$ be a group with subgroups $H$ and $K$. If for every $g \in G$ and $k \in K$ we know that $g^{-1}kg \in K$, prove that $P$ is a subgroup of $G$, where $P = \{hk \mid h \in H, k \in K\}$.
I already proved the closure property but when I try to prove the inverse property, I end up with $ha^{-1} = hk^{-1}h^{-1}$!

REPLY [1 votes]: So $K\triangleleft G$.  Now $(hk)^{-1}=k^{-1}h^{-1}=h^{-1}hk^{-1}h^{-1}=h^{-1}k'\in P$.
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