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The class equation $∣ G ∣ = ∣ Z (G) ∣ + \sum_{i} [G : C_{G} (x_{i})]$ for a finite group

Statement

Let $G$ be finite, and let $x_{1}, \dots, x_{r}$ contain one representative from each conjugacy class having more than one element. Then

$∣ G ∣ = ∣ Z (G) ∣ + \sum_{i = 1}^{r} [G : C_{G} (x_{i})] .$

Facts & Assumptions

Given: A finite group $G$ and representatives $x_{1}, \dots, x_{r}$ of its non-singleton conjugacy classes.

[L1]

The orbits of an action partition the acted-on set (The orbits of a group action are the equivalence classes of $x \sim y$ iff $y = g \cdot x$ for some $g$ , and hence partition the acted-on set).

[L2]

Under conjugation, $∣ {Cl}_{G} (x) ∣ = [G : C_{G} (x)]$ ( $G / C_{G} (x) \to {Cl}_{G} (x)$ is a bijection, so $∣ {Cl}_{G} (x) ∣ = [G : C_{G} (x)]$ whenever these cardinalities are finite).

[L3]

Conjugacy classes and centralizers are as in The conjugacy class ${Cl}_{G} (x)$ and centralizer $C_{G} (x)$ of an element.

[L4]

The center is $Z (G) = {z \in G : z g = g z for every g \in G}$ (The center $Z (G)$ of a group).

[L5]

A finite partition has cardinality equal to the sum of its block cardinalities (The sum rule: a finite disjoint union is finite with $∣ A \cup B ∣ = ∣ A ∣ + ∣ B ∣$ and $∣ ⋃_{i \in I} A_{i} ∣ = \sum_{i \in I} ∣ A_{i} ∣$ , and a sum over a finite index set splits along a partition).

[L6]

Finite sums over finite index sets are well-defined (The sum $\sum_{i \in S} a_{i}$ over a finite index set, and its product form).

Proof

technique · direct

1.1

Let $G$ act on itself by conjugation. By [L1] and [L3], its orbits are the conjugacy classes and they partition $G$ .

L1L3

2.1

The class of $x$ is a singleton exactly when $g x g^{- 1} = x$ for every $g$ , equivalently when $x \in Z (G)$ by [L4]. Thus the singleton classes contribute $∣ Z (G) ∣$ .

step 1.1L3L4

3.1

Applying the finite partition sum rule to the singleton classes and to the classes represented by $x_{1}, \dots, x_{r}$ gives $∣ G ∣ = ∣ Z (G) ∣ + \sum_{i = 1}^{r} ∣ {Cl}_{G} (x_{i}) ∣$ .

step 1.1step 2.1L5L6

4.1

Replacing each remaining class size by [L2] yields $∣ G ∣ = ∣ Z (G) ∣ + \sum_{i = 1}^{r} [G : C_{G} (x_{i})]$ .

step 3.1L2∎

Depends on

Used by

Dependency tree · two levels

33 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

T. W. Judson, Abstract Algebra: Theory and Applications, 14.2, The Class Equation (standard reference, not scraped)

The class equation ∣G∣=∣Z(G)∣+∑i[G:CG(xi)]∣G∣=∣Z(G)∣+∑i​[G:CG​(xi​)] for a finite group