Definition:Conjugate (Group Theory)

Definition

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

Conjugate of an Element

An element $x \in G$ is conjugate to an element $y \in G$ iff:

$\exists a \in G: a \circ x = y \circ a$

Alternatively, we can say that $x$ is the conjugate of $y$ by $a$.

This relation is called conjugacy.

We write $x \sim y$ for $x$ is a conjugate of $y$.

This relation is alternatively (and usually) expressed as:

$x \sim y := a \circ x \circ a^{-1} = y$

which is seen to be equivalent to the other definition by taking the group product on the right with $a^{-1}$.

Conjugate of a Set

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

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

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

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 (as is usual), we usually just refer to the conjugate of $S$ by $a$.