Conjugacy class

In mathematics, especially group theory, two elements ${\displaystyle a}$ and ${\displaystyle b}$ of a group are conjugate if there is an element ${\displaystyle g}$ in the group such that ${\displaystyle b=gag^{-1}.}$ This is an equivalence relation whose equivalence classes are called conjugacy classes. In other words, each conjugacy class is closed under ${\displaystyle b=gag^{-1}}$ for all elements ${\displaystyle g}$ in the group.

Members of the same conjugacy class cannot be distinguished by using only the group structure, and therefore share many properties. The study of conjugacy classes of non-abelian groups is fundamental for the study of their structure. For an abelian group, each conjugacy class is a set containing one element (singleton set).

Functions that are constant for members of the same conjugacy class are called class functions.

The concept of conjugacy classes may come from trying to formalize the idea that two group elements are considered the "same" after a relabeling of elements.

For example, consider the symmetric group ${\displaystyle S_{5}}$ of order 5, and elements ${\displaystyle \sigma }$ and ${\displaystyle \pi \sigma \pi ^{-1}}$ that are conjugate. An element ${\displaystyle \pi \sigma \pi ^{-1}}$ can be viewed as simply "renaming" the elements ${\displaystyle 1,2,3,4,5}$ to ${\displaystyle \pi (1),\pi (2),\pi (3),\pi (4),\pi (5)}$ then applying the permutation ${\displaystyle \sigma }$ on this new labeling.

The conjugacy action by ${\displaystyle \pi }$ does not change the underlying structure of ${\displaystyle \sigma }$. In a way, permutations ${\displaystyle \sigma }$ and ${\displaystyle \pi \sigma \pi ^{-1}}$ have the same "shape".

Another way to view the conjugacy action is by considering the general linear group ${\displaystyle \operatorname {GL} (n)}$ of invertible matrices. Two matrices ${\displaystyle A}$ and ${\displaystyle B}$ conjugate if there exist a matrix ${\displaystyle P}$ such that ${\displaystyle B=PAP^{-1}}$, which is the same condition as matrix similarity. The two matrices are conjugates if they are the "same" under two possibly different bases, with ${\displaystyle P}$ being the change-of-basis matrix.

Conjugates also come up in some important theorems of group theory. One example is the Sylow theorems, which state that every Sylow ${\displaystyle p}$-subgroup of a finite group ${\displaystyle G}$ are conjugates to each other. It also appears in the proof of Cauchy's theorem, which makes use of conjugacy classes.