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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 $x_1,\ldots,x_r$ 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 $x_1,\ldots,x_r$ 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, $|\operatorname{Cl}_G(x)|=[G:C_G(x)]$ ( $G/C_G(x)\to\operatorname{Cl}_G(x)$ is a bijection, so $|\operatorname{Cl}_G(x)|=[G:C_G(x)]$ whenever these cardinalities are finite).

[L3]

Conjugacy classes and centralizers are as in The conjugacy class $\operatorname{Cl}_G(x)$ and centralizer $C_G(x)$ of an element.

[L4]

The center is $Z(G)=\{z\in G:zg=gz\text{ for every }g\in G\}$ (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 $\lvert A \cup B\rvert = \lvert A\rvert + \lvert B\rvert$ and $\lvert\bigcup_{i \in I} A_i\rvert = \sum_{i \in I}\lvert A_i\rvert$ , 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 $gxg^{-1}=x$ 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 $x_1,\ldots,x_r$ gives $|G|=|Z(G)|+\sum_{i=1}^{r}|\operatorname{Cl}_G(x_i)|$ .

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

The class equation of S₃ is 6=1+2+3 Example
The square-symmetry group has class equation 8=2+2+2+2 Example

Dependency tree · next 3 levels

Direct dependencies and their dependencies through the next three levels: 97 results over 21 levels. An arrow runs from a result to what uses it, and this result sits at the bottom with a heavier outline. Click 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)|+\sum_i [G:C_G(x_i)]∣G∣=∣Z(G)∣+∑i​[G:CG​(xi​)] for a finite group