Group Acts on Itself

Theorem

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

Then $\struct {G, \circ}$ acts on itself by the rule:

$\forall g, h \in G: g * h = g \circ h$

Proof

Follows directly from the group axioms and the definition of a group action.

$\blacksquare$

Also see

• Stabilizer of Element of Group Acting on Itself is Trivial
• Orbit of Element of Group Acting on Itself is Group

Sources

• 1971: Allan Clark: Elements of Abstract Algebra ... (previous) ... (next): Chapter $2$: The Sylow Theorems: $\S 53$
• 1996: John F. Humphreys: A Course in Group Theory ... (previous) ... (next): Chapter $10$: The Orbit-Stabiliser Theorem: Example $10.4$