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# 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 $g \in G$, and let $c_g \colon G \rightarrow G$ denote conjugation by $g$
First, we check that $c_g$ is a group homomorphism.
 Suppose that $x,y \in G$. Then $c_g(xy) = g xy g^{-1} = g x g^{-1} g y g^{-1} = c_g(x) c_g(y)$, using the associativity of composition. Hence $c_g$ is a group homomorphism from $G$ to itself.

Next, we check that $c_{g^{-1} }$ is an inverse function of $c_g$:

 Suppose that $x \in G$. Then we can compute: $$c_g c_{g^{-1} }(x) = g (g^{-1} x (g^{-1})^{-1}) g^{-1} = g g^{-1} x g g^{-1} = x.$$ Replacing $g$ by $g^{-1}$ in the above computation, we find that $c_{g^{-1} } c_g(x) = x$. Hence $c_{g^{-1} } \circ c_g = c_g \circ c_{g^{-1} } = Id_G$.

## Logical Connections

This definition logically relies on the following definitions and statements: Def/Group, Def/Group automorphism, Def/Binary relation, Def/Equivalence relation, State/Homomorphic images of subgroups are subgroups

The following statements and definitions logically rely on the material of this page: State/Centers of p-groups are nontrivial, and State/Conjugacy of Sylow subgroups

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