Definition:Conjugacy Action

Definition

Let $\struct {G, \circ}$ be a group.

The (left) conjugacy action of $G$ is the left group action $* : G \times G \to G$ defined as:

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

The right conjugacy action of $G$ is the right group action $* : G \times G \to G$ defined as:

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

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

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

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

The right conjugacy action on subgroups is the group action $* : X \times G \to X$:

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

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 $* : \powerset G \times G \to \powerset G$:

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

Some sources refer to this group action of $g * x$ as the transform of $x$ by $g$.

1965: J.A. Green: Sets and Groups ... (previous) ... (next): $\S 5.5$. Groups acting on sets: Example $103$
1982: P.M. Cohn: Algebra Volume 1 (2nd ed.) ... (previous) ... (next): $\S 3.3$: Group actions and coset decompositions: Examples of group actions: $\text{(v)}$
1996: John F. Humphreys: A Course in Group Theory ... (previous) ... (next): Chapter $10$: The Orbit-Stabiliser Theorem: Example $10.5$