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:
 * $$x \sim y \ \stackrel {\mathbf {def}} {=\!=} \ a \circ x \circ a^{-1} = y$$

which is seen to be equlvalent 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:
 * $$x \sim y \ \stackrel {\mathbf {def}} {=\!=} \ \exists a \in G: x \circ a = a \circ y$$

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

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:


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

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:


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

There is a subtle difference between the definitions.

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