Orbit of Conjugacy Action on Subgroup is Set of Conjugate Subgroups

Theorem

Let $\struct {G, \circ}$ be a group whose identity is $e$.

Let $X$ be the set of all subgroups of $G$.

Let $*$ be the conjugacy action on $H$ defined as:

$\forall g \in G, H \in X: g * H = g \circ H \circ g^{-1}$

Then the orbit $\Orb H$ of $H$ in $\powerset G$ is the set of subgroups of $G$ conjugate to $H$.

Proof

We have that:

$\Orb H = \set {g \circ H \circ g^{-1}: g \in G}$

from the definition.

The result follows by definition of conjugate subgroup.

$\blacksquare$