Conjugacy Classes: Definition and Computation

1. Conjugacy: A New Equivalence Relation

At its core, conjugation is a relationship between elements mediated by other elements of the group.

Definition: In a group $G$, two elements $a$ and $b$ are said to be conjugate if there exists an element $g\in G$ such that:

In this case, we say that $g$ conjugates $a$ into $b$.

This relationship is an equivalence relation, meaning it is:

1. Reflexive: Any element $a$ is conjugate to itself (choose $g=e$, the identity).
2. Symmetric: If $b$ is conjugate to $a$ (i.e., $b=ga{g}^{-1}$), then $a$ is conjugate to $b$ (since $a=g^{-1}b(g^{-1}{)}^{-1}=g^{-1}bg$).
3. Transitive: If $b$ is conjugate to $a$ and $c$ is conjugate to $b$, then $c$ is conjugate to $a$.

Because conjugacy is an equivalence relation, it partitions the group $G$ into disjoint sets called conjugacy classes. The conjugacy class of an element $a$, denoted $K_a$, is the set of all elements in $G$ that are conjugate to $a$.

To build a solid foundation, let's turn to a formal reading that defines these concepts and proves some of their most important initial properties.

The analogy to linear algebra, where similar matrices ($B=PA{P}^{-1}$) share properties like eigenvalues and trace, is particularly apt. The following video elaborates on this connection.

2. The Class Equation: A Tool for Counting

We now have a picture of a group partitioned into conjugacy classes. A natural question follows: what determines the size of each class? The answer lies in another important substructure: the centralizer.

Definition: The centralizer of an element $a\in G$, denoted $Z(a)$ or $C_G(a)$, is the set of all elements in $G$ that commute with $a$:

It is straightforward to verify that $Z(a)$ is always a subgroup of $G$. The centralizer should not be confused with the center of the group, $Z(G)$, which is the set of elements that commute with all elements in $G$. The center is the intersection of all centralizers: $Z(G)={\bigcap }_{a\in G}Z(a)$.

The size of a conjugacy class is directly related to the size of the centralizer of any of its elements. The following theorem makes this connection precise.

Theorem: The size of the conjugacy class of $a$, denoted $|K_a|$, is equal to the index of its centralizer subgroup, $[G:Z(a)]$.

This is a powerful result. By Lagrange's theorem, it immediately implies that the size of every conjugacy class must be a divisor of the order of the group.

Summing the sizes of all the disjoint conjugacy classes must yield the total order of the group. This gives us the class equation:

where the sum is over a set of representative elements $a_i$ from each distinct conjugacy class. It is often useful to separate the terms for the center of the group, since for any element $a\in Z(G)$, its class size is 1. The equation can then be written as:

where the sum now runs over representatives of classes with more than one element.

Let's watch a short segment that clearly defines the centralizer.

For a rigorous proof of the relationship between the class size and the centralizer, we return to Conrad's notes.

3. Computing Conjugacy Classes: Examples

With these tools, we can now systematically determine the conjugacy classes for some important finite groups.

Example 1: The Quaternion Group, $Q_8$

The quaternion group $Q_8=\{\pm 1,\pm i,\pm j,\pm k\}$ is a non-Abelian group of order 8 with multiplication rules $i^2=j^2=k^2=ijk=-1$. This group is intimately related to the Lie group SU(2), which describes spin in quantum mechanics and rotations. A group generated by the Pauli matrices is closely related to $Q_8$, a connection you can explore in Problem 15 of the tutorial sheet from Universität Bielefeld.

Let's compute the conjugacy class of the element $i$.

• Center: First, we identify the center $Z(Q_8)$. The elements $1$ and $-1$ commute with all other elements, so $Z(Q_8)=\{1,-1\}$. These each form a conjugacy class of size 1: $K_1=\{1\}$ and $K_{-1}=\{-1\}$.
• Centralizer of $i$: We find the elements that commute with $i$. By inspection or direct calculation ($ji=-k\ne k=ij$), we find that $j$ and $k$ do not commute with $i$. The elements that do commute are $\{1,-1,i,-i\}$. Thus, $Z(i)=\{1,-1,i,-i\}$.
• Class Size: The size of the centralizer is $|Z(i)|=4$. Using our theorem, the size of the conjugacy class of $i$ is $|K_i|=|Q_8|/|Z(i)|=8/4=2$.
• The Class: We know $i$ is in its own class. We need to find one other element. Let's conjugate $i$ by $j$: $$ji{j}^{-1}=ji(-j)=(-k)(-j)=kj=-i$$ So, $-i$ is in the class of $i$. Since we know the class has exactly two elements, we are done: $K_i=\{i,-i\}$.
• By symmetry, the other classes are $K_j=\{j,-j\}$ and $K_k=\{k,-k\}$.

The class equation for $Q_8$ is $8=|K_1|+|K_{-1}|+|K_i|+|K_j|+|K_k|=1+1+2+2+2$.

Example 2: The Dihedral Group, $D_4$

The group $D_4$ is the group of symmetries of a square, with 8 elements: four rotations ($e,r,r^2,r^3$) and four reflections ($s,rs,r^{2}s,r^{3}s$). This is a classic example in physics. The following video provides a detailed, step-by-step computation of its five conjugacy classes.

The resulting classes for $D_4$ are: $\{e\}$, $\{r^2\}$, $\{r,r^3\}$, $\{s,r^{2}s\}$, and $\{rs,r^{3}s\}$. The class equation is $8=1+1+2+2+2$.

Example 3: The Symmetric Group, $S_n$

For the symmetric groups $S_n$, there is a remarkably elegant result: two permutations are conjugate if and only if they have the same cycle structure.

The cycle structure (or cycle type) of a permutation is the list of the lengths of its disjoint cycles, including 1-cycles for fixed points. For example, in $S_5$, the permutation $(12)(345)$ has cycle structure (2, 3). The permutation $(12)(45)$ leaves 3 fixed, so its disjoint cycle decomposition is $(3)(12)(45)$, and its cycle structure is (1, 2, 2).

This theorem means that to find the conjugacy classes of $S_n$, we simply need to find all possible ways to partition the integer $n$. For $S_4$, the partitions of 4 are:

• 1+1+1+1 $\to$ The identity $(e)$. Class size: 1.
• 2+1+1 $\to$ Transpositions like $(12)$. Class size: 6.
• 2+2 $\to$ Products of two disjoint transpositions like $(12)(34)$. Class size: 3.
• 3+1 $\to$ 3-cycles like $(123)$. Class size: 8.
• 4 $\to$ 4-cycles like $(1234)$. Class size: 6.

The class equation is $24=1+6+3+8+6$. The following reading formalizes this powerful theorem.

This principle extends to continuous groups as well. For the rotation group SO(3), which is central to your study of quantum mechanics and field theory, all rotations by the same angle $\theta$, regardless of the axis of rotation, form a single conjugacy class.

Conclusion

In this lesson, we have introduced a powerful new way to analyze the internal structure of a group. By partitioning a group into its conjugacy classes, we group together elements that are algebraically and geometrically "of the same type."

Here are the main points we have covered:

• Two elements $a,b$ are conjugate if $b=ga{g}^{-1}$ for some $g\in G$. This defines an equivalence relation that partitions the group into disjoint conjugacy classes.
• All elements within a single conjugacy class have the same order.
• The centralizer $Z(a)$ is the subgroup of elements that commute with $a$.
• The size of a conjugacy class is given by the index of the centralizer: $|K_a|=|G|/|Z(a)|$. This implies the size of any class must divide the order of the group.
• The class equation, $|G|=\sum |K_{a_i}|$, provides a fundamental constraint on the structure of a finite group.
• We learned how to compute conjugacy classes for specific groups like $Q_8$ and $D_4$, and saw that for $S_n$, conjugacy is determined by cycle structure.

In our last lesson, we partitioned $G$ into cosets of a subgroup $H$. In this lesson, we partitioned $G$ into conjugacy classes. What is the connection? A particularly important class of subgroups are those that are themselves a union of entire conjugacy classes. Such subgroups are called normal subgroups, and they are precisely the subgroups for which the set of cosets can be endowed with a group structure of its own. In our next lesson, we will formally define normal subgroups and begin constructing these new "quotient groups."