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Let $G$ be a closed, connected Lie group, and let $H$ be a closed (and therefore Lie) subgroup. There is a natural action of $G$ on the space of left cosets $G/H$, for which the stabiliser of $aH$ is the conjugate subgroup ${}^aH:=aHa^{-1}$.

Now let $G$ act diagonally on $G/H\times G/H$. The stabiliser of $(aH,bH)$ is the intersection ${}^aH\cap {}^bH$ of conjugate subgroups. My question is, what can be said about ${}^aH\cap {}^bH$?

I know that ${}^aH={}^bH$ if and only if $ab^{-1}\in N(H)$, the normaliser of $H$ in $G$. Beyond this I couldn't find much via Google. For instance can ${}^aH\cap {}^bH$ be trivial? Or must it be a conjugate of $H$?

Apologies if this is too elementary.

Edit: Thanks for the answers, which show that not much can be said at this level of generality. Now I am looking at a specific example - the icosahedral group $I\cong A_5$ inside $SO(3)$ (where the space of left cosets is the PoincarĂ© sphere). This subgroup has the properties that $I$ is finite, $N(I)=I$ and $I/[I,I]$ is trivial. Does this allow me to conclude that conjugates of $I$ are either equal or intersect in the identity?

More generally, what can be said if I add the assumption $N(H)=H$ to the original question?

Edit 2: I've now asked about the icosahedral group in another question.

Let $G$ be a closed, connected Lie group, and let $H$ be a closed (and therefore Lie) subgroup. There is a natural action of $G$ on the space of left cosets $G/H$, for which the stabiliser of $aH$ is the conjugate subgroup ${}^aH:=aHa^{-1}$.

Now let $G$ act diagonally on $G/H\times G/H$. The stabiliser of $(aH,bH)$ is the intersection ${}^aH\cap {}^bH$ of conjugate subgroups. My question is, what can be said about ${}^aH\cap {}^bH$?

I know that ${}^aH={}^bH$ if and only if $ab^{-1}\in N(H)$, the normaliser of $H$ in $G$. Beyond this I couldn't find much via Google. For instance can ${}^aH\cap {}^bH$ be trivial? Or must it be a conjugate of $H$?

Apologies if this is too elementary.

Edit: Thanks for the answers, which show that not much can be said at this level of generality. Now I am looking at a specific example - the icosahedral group $I\cong A_5$ inside $SO(3)$ (where the space of left cosets is the PoincarĂ© sphere). This subgroup has the properties that $I$ is finite, $N(I)=I$ and $I/[I,I]$ is trivial. Does this allow me to conclude that conjugates of $I$ are either equal or intersect in the identity?

More generally, what can be said if I add the assumption $N(H)=H$ to the original question?

3 typo

Let $G$ be a closed, connected Lie group, and let $H$ be a closed (and therefore Liesubgroup)) subgroup. There is a natural action of $G$ on the space of left cosets $G/H$, for which the stabiliser of $aH$ is the conjugate subgroup ${}^aH:=aHa^{-1}$.

Now let $G$ act diagonally on $G/H\times G/H$. The stabiliser of $(aH,bH)$ is the intersection ${}^aH\cap {}^bH$ of conjugate subgroups. My question is, what can be said about ${}^aH\cap {}^bH$?

I know that ${}^aH={}^bH$ if and only if $ab^{-1}\in N(H)$, the normaliser of $H$ in $G$. Beyond this I couldn't find much via Google. For instance can ${}^aH\cap {}^bH$ be trivial? Or must it be a conjugate of $H$?

Apologies if this is too elementary.

2 typos
1