Let $H = \{e, (12)\} \subset S_3$. Find the orbit of $H$ under conjugation by $G$.

Question

What I did was just using the definitions.

The orbit of $x$ in $G$ is $O_G(x) = \{ghg^{-1} : g \in G\}$

So, for $x = e$ we get $O_{S_3}(e) = \{geg^{-1}:g\in S_3\} = \{ g g^{-1} : g \in S_3\} = \{e\}$

WLOG, for $x = (12)$ I got

$O_{S_3}((12)) = \{(12), (13), (23)\}$

My problem is, these are the individual orbits of each element of $H$. What is the orbit of $H$ itself? Is it the set of all the orbits for each $x \in H$, in this case $\{\{e\}, \{(12), (13), (23)\}\}$?

Answer 1

The orbit of $H$ itself is the set $\{gHg^{-1}\mid g\in S_3\}$ of subgroups. Explicitly, it is $\{H, \{e, (13)\}, \{e, (23)\}\}$.