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 obtaining the group product on the left with $a^{-1}$.

Alternative Definition
There is an alternative way of defining conjugacy of elements, which is subtly different:

$x$ is a conjugate of $y$ iff:
 * $x \sim y := \exists a \in G: x \circ a = a \circ y$

or:
 * $x \sim y := \exists a \in G: a^{-1} \circ x \circ a = y$

This is clearly equivalent to the other definition by noting that if $a \in G$ then $a^{-1} \in G$ also.

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 $a$.

Alternative Definition for Set
Similarly to the alternative definition for group elements, the concept of set conjugacy can be defined as:


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

There is a subtle difference between the definitions.

See, for example, Conjugate of a Set by Product.

Alternative terminology
Some sources call $a \circ x \circ a^{-1}$ (or $a^{-1} \circ x \circ a$) the transform of $x$ by $a$.

Also see

 * Conjugacy is an Equivalence