---
kind: statement
name: Stabiliser sizes over an orbit add up to the order of the group
role: proposition
msc:
  - "20B05"
  - "05E18"
sources:
  - url: https://github.com/mitchkeller/applied-combinatorics/blob/ebd95d6bd59e23c633c65cf560bcb93efa912aae/source/s_polya_burnside.ptx#L9
    locator: source/s_polya_burnside.ptx:9
tag: CQTDS8B6R7
uses:
  - G5S1M9
  - 0WCY42EV9V
  - 0YQPZB5Q1G
  - 0AYMP41THN
  - 0E2HSCKZ6C
---

## Statement

Let a [[group|G5S1M9]] $G$ [[act|0WCY42EV9V]] on a finite set $X$, and let $x \in X$.
Write $G \cdot x$ for the [[orbit|0YQPZB5Q1G]] of $x$, and $G_y$ for the [[stabiliser|0AYMP41THN]] of a point $y$.
Then the sizes of the stabilisers of all points of the orbit add up to the order of the group:

$$\sum_{y \in G \cdot x} |G_y| = |G|.$$

## Remarks

The stabilisers of the points of one orbit are conjugate, so they all have the same size.
By the [[orbit-stabiliser theorem|0E2HSCKZ6C]], this common size is $|G| / |G \cdot x|$.
The orbit has $|G \cdot x|$ terms, so the sum is $|G|$.

For example, let $G = \mathbb{Z}/4\mathbb{Z}$ rotate the corners of a square. The orbit of a corner has $4$ points, each with trivial stabiliser, so the sum is $4 = |G|$.