# 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:

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

### Conjugacy Action on Subgroups

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 $* : X \times G \to X$:

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

### Conjugacy Action on Subsets

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 known as

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

## Also see

• Results about the conjugacy action can be found here.