Definition:Conjugacy Action/Subsets

Definition

Let $G$ be a group.

Let $\powerset G$ be the power set of $G$.

The (left) conjugacy action on subsets is the group action $* : G \times \powerset G \to \powerset G$:

$g * S = g \circ S \circ g^{-1}$

The right conjugacy action on subsets is the group action $* : \powerset G \times G \to \powerset G$:

$S * g = g^{-1} \circ S \circ g$

Also see

• Conjugacy Action is Group Action

Sources

• 1965: J.A. Green: Sets and Groups ... (previous) ... (next): $\S 5.5$. Groups acting on sets: Example $105$
• 1971: Allan Clark: Elements of Abstract Algebra ... (previous) ... (next): Chapter $2$: The Sylow Theorems: $\S 54$