While going through my class notes, I came across a statement I copied down from the board that I don't quite understand. The statement is this:
Let $H$ be a subgroup of a group $G$. Suppose that $g \in G$ but $ g \notin H$, and that $g$ has order $2$ in $G$, and moreover $gHg^{-1} = H$. Then if $|H| = n$, the subgroup of $G$ generated by $H$ and $g$ has size $2n$.
I'm trying to do this on my own by showing that the only cosets of $H$ in $\langle g,H \rangle$ are $H$ and $gH$, and I guess the condition that $g$ has order $2$ tells us we don't get any extra cosets of the form $g^nH$, but I'm not seeing how to use the condition that $g$ conjugates $H$ into itself to show that there are no other cosets. Any help is appreciated!