Conjugate Element

Given a Group with elements $A$ and $X$, there must be an element $B$ which is a Similarity Transformation of $A, B = X^{-1} AX$ so $A$ and $B$ are conjugate with respect to $X$. Conjugate elements have the following properties:

1. Every element is conjugate with itself.

2. If $A$ is conjugate with $B$ with respect to $X$, then $B$ is conjugate to $A$ with respect to $X$.

3. If $A$ is conjugate with $B$ and $C$, then $B$ and $C$ are conjugate with each other.

See also Conjugacy Class, Conjugate Subgroup