Definition:Conjugate (Group Theory)

Definition

Let $\left({G, \circ}\right)$ be a group.

Conjugate of an Element

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

Definition 1

The conjugacy relation $\sim$ is defined on $G$ as:

$\forall \tuple {x, y} \in G \times G: x \sim y \iff \exists a \in G: a \circ x = y \circ a$

Definition 2

The conjugacy relation $\sim$ is defined on $G$ as:

$\forall \tuple {x, y} \in G \times G: x \sim y \iff \exists a \in G: a \circ x \circ a^{-1} = y$

Conjugate of a Set

Let $S \subseteq G, a \in G$.

Then the $G$-conjugate of $S$ by $a$ is:

$S^a := \set {y \in G: \exists x \in S: y = a \circ x \circ a^{-1} } = a \circ S \circ a^{-1}$

That is, $S^a$ is the set of all elements of $G$ that are the conjugates of elements of $S$ by $a$.

When $G$ is the only group under consideration, we usually just refer to the conjugate of $S$ by $a$.