Conjugacy Action on Subgroups is Group Action

Theorem

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

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

For any $H \le G$ and for any $g \in G$, the conjugacy action:

$g * H := g \circ H \circ g^{-1}$

is a group action.

Proof

Clearly Group Action Axiom $\text {GA} 1$ is fulfilled as $e * H = H$.

Group Action Axiom $\text {GA} 2$ is shown to be fulfilled thus:

\(\ds \paren {g_1 \circ g_2} * H\)

\(=\)

\(\ds \paren {g_1 \circ g_2} \circ H \circ \paren {g_1 \circ g_2}^{-1}\)

Definition of $*$

\(\ds \)

\(=\)

\(\ds g_1 \circ g_2 \circ H \circ g_2^{-1} \circ g_1^{-1}\)

Inverse of Group Product

\(\ds \)

\(=\)

\(\ds g_1 * \paren {g_2 \circ H \circ g_2^{-1} }\)

Definition of $*$

\(\ds \)

\(=\)

\(\ds g_1 * \paren {g_2 * H}\)

Definition of $*$

$\blacksquare$

Also see

• Stabilizer of Conjugacy Action on Subgroup is Normalizer
• Orbit of Conjugacy Action on Subgroup is Set of Conjugate Subgroups

Sources

• 1996: John F. Humphreys: A Course in Group Theory ... (previous) ... (next): Chapter $10$: The Orbit-Stabiliser Theorem: Example $10.7$