Def/Conjugate

Definition of Conjugate: Suppose that $G$ is a group. The words conjugate and conjugacy are used in numerous similar situations, which are covered in sections here.

The most fundamental notion of conjugating is the following: suppose that $g \in G$. Then, there is a function $c_g \colon G \rightarrow G$, called "conjugation by $g$", defined as follows: $$c_g(x) = g x g^{-1}, \mbox{ for all } x \in G.$$ The element $g x g^{-1}$ is called the conjugate of $x$ by $g$. The most important property of conjugation is that it is an automorphism of $G$:

Suppose that $x$ and $y$ are elements of $G$. We say that $x$ is conjugate to $y$, if there exists $g \in G$, such that $g x g^{-1} = y$. This defines a relation on $G$, called conjugacy. Conjugacy is an equivalence relation:

Since conjugacy is an equivalence relation, it partitions $G$ into equivalence classes, which are called conjugacy classes.

Suppose that $H$ is a subgroup of $G$, and $g \in G$. Define a subset $g H g^{-1} \subset G$ by: $$g H g^{-1} = c_g(H) = \{ x \in G \mbox{ such that } \exists h \in H, x = g h g^{-1} \}.$$ Then, $g H g^{-1} = c_g(H)$ is a subgroup of $G$, since homomorphic images of subgroups are subgroups. Furthermore, since $c_g$ is an automorphism of $G$, $c_g$ restricts to an isomorphism of groups from $H$ to $g H g^{-1}$.