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Expert ∞ Differential Equations, Abstract Algebra & Applied Math Topic 76 of 97

Orbit–Stabiliser Theorem

Orbit size times stabiliser size equals group order.

Key idea

Orbit size times stabiliser size equals group order.

In depth

For a group acting on X, |Orbit(x)| × |Stab(x)| = |G|. It links the size of the orbit with the symmetry that leaves x fixed, and is used to count symmetries.

Worked example

The rotation group of the cube has order 24; a vertex has orbit size 8 and stabiliser size 3.

Formula

|G| = |Orb||Stab|

Real-world uses

Common mistake

Forgetting that the theorem concerns a single point.

How to learn this

  1. Read the key idea and say it in your own words.
  2. Work through the worked example, then try a similar problem.
  3. Use the interactive lesson’s visual and mini-quiz to check the pattern.

Frequently asked questions

What is Orbit–Stabiliser Theorem?

Orbit size times stabiliser size equals group order. It is a topic at the expert level in Differential Equations, Abstract Algebra & Applied Math.

What is the formula for Orbit–Stabiliser Theorem?

The key formula is |G| = |Orb||Stab|.

Where is Orbit–Stabiliser Theorem used in real life?

Real-world examples: Counting the different positions of a molecule or the arrangements of a puzzle.

What should I learn before Orbit–Stabiliser Theorem?

Make sure you are comfortable with Group Actions, the previous topic in Differential Equations, Abstract Algebra & Applied Math.

What is a common mistake with Orbit–Stabiliser Theorem?

Forgetting that the theorem concerns a single point.

How can I practise Orbit–Stabiliser Theorem?

Open the interactive lesson, play with the visual, then answer the three-question mini-quiz. Getting two right marks the topic complete.

What comes after Orbit–Stabiliser Theorem?

Move on to Burnside’s Lemma next.

Related expert topics in Differential Equations, Abstract Algebra & Applied Math