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The Class Equation

Let X be a finite G-set and X_G be the set of fixed points in X; that is, \[\begin{aligned}\end{aligned}\].

The Class Equation

Let \(X\) be a finite \(G\)-set and \(X_G\) be the set of fixed points in \(X\); that is, \[\begin{aligned}\end{aligned}\]. Since the orbits of the action partition \(X\), \[\begin{aligned}\end{aligned}\], where \(x_k, \ldots, x_n\) are representatives from the distinct nontrivial orbits of \(X\).

Now consider the special case in which \(G\) acts on itself by conjugation, \((g,x) \mapsto gxg^{-1}\). The center of \(G\), \[\begin{aligned}\end{aligned}\], is the set of points that are fixed by conjugation. The nontrivial orbits of the action are called the conjugacy classes of \(G\). If \(x_1, \ldots, x_k\) are representatives from each of the nontrivial conjugacy classes of \(G\) and \(|{\mathcal O}_{x_1}| = n_1, \ldots, |{\mathcal O}_{x_k}| = n_k\), then \[\begin{aligned}\end{aligned}\]. The stabilizer subgroups of each of the \(x_i\)'s, \(C(x_i) = \{ g \in G: g x_i = x_i g \}\), are called the centralizer subgroups of the \(x_i\)'s. From , we obtain the class equation: \[\begin{aligned}\end{aligned}\]. One of the consequences of the class equation is that the order of each conjugacy class must divide the order of \(G\).

Example

It is easy to check that the conjugacy classes in \(S_3\) are the following: \[\begin{aligned}\end{aligned}\]. The class equation is \(6 = 1+2+3\).

Example

The center of \(D_4\) is \(\{ (1), (1 \, 3)(2 \, 4) \}\), and the conjugacy classes are \[\begin{aligned}\end{aligned}\]. Thus, the class equation for \(D_4\) is \(8 = 2 + 2 + 2 + 2\).

Condensed — the full section is in Judson, Abstract Algebra: Theory and Applications.

Symbols used here

a \mid b,\ \gcd(a,b)
divides, greatest common divisor
b is a multiple of a; the largest number dividing both.
x \in A,\ A \subseteq B
element of, subset
x belongs to A; every element of A is in B.
\leq,\ \geq
less/greater than or equal
Inequalities that allow equality; < and > exclude it.
\neq
not equal
The two sides are different.
\mathbb{N},\ \mathbb{Z},\ \mathbb{Q},\ \mathbb{R},\ \mathbb{C}
number sets
Naturals, integers, rationals, reals, complex numbers.
\blacksquare\ \text{or}\ \square
end of proof (halmos)
Marks the point where the statement has been established.
a \equiv b \pmod n
congruent modulo n
n divides a − b; a and b have the same remainder.
(G, \cdot),\ e,\ g^{-1}
group, identity, inverse
A set with an operation; the do-nothing element; the element that undoes g.
G \cong H,\ G / N
isomorphic, quotient group
Same structure; the group of cosets of a normal subgroup N.
\mathbb{Z}/n\mathbb{Z},\ \mathbb{Z}_n
integers modulo n
The remainders 0…n−1 with clock arithmetic.
\operatorname{Hom}(A, B),\ f \circ g
arrows from A to B, composition
The set of morphisms; do g then f.

Questions people ask

What is a group, in plain words?

A set with one operation that is associative, has an identity, and lets every element be undone. Symmetries of any object form a group — that is where the idea came from.

What is the difference between a ring and a field?

A ring has addition and multiplication that behave like the integers (you cannot always divide); a field is a ring where every non-zero element has a reciprocal, like the rationals or the reals.

Ipprova tiegħek stess

Parts of this page are adapted from Judson, Abstract Algebra: Theory and Applications (GFDL 1.3). Condensed and re-explained here; errors are ours.

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