Orbit-Stabilizer Theorem Calculator

Formula first

Overview

The Orbit-Stabilizer Theorem establishes a fundamental relationship between a group acting on a set and the symmetry of the elements within that set. It states that the size of the group is equal to the product of the size of an element's orbit and the order of its stabilizer subgroup.

Symbols

Variables

|G| = |G|

Apply it well

When To Use

When to use: Use this theorem when you need to calculate the number of unique arrangements under symmetry or determine the size of a symmetry group. It is applicable whenever a finite group G acts on a finite set X.

Why it matters: This theorem is the cornerstone of group theory applications in combinatorics, chemistry (molecular symmetry), and crystallography. It allows mathematicians to simplify complex counting problems by focusing on fixed points and stabilizers.

Avoid these traps

Common Mistakes

• Confusing the size of the set X with the size of the orbit of a specific element.
• Assuming all elements in the set have the same orbit size.
• Mistaking the stabilizer for the centralizer or other subgroups.

One free problem

Practice Problem

A group G of order 24 acts on a set X. If the stabilizer of an element x has exactly 4 elements, what is the size of the orbit of x?

Solve for: $|G|$

Hint: The product of the orbit size and the stabilizer size equals the group order.

The full worked solution stays in the interactive walkthrough.

Keep going

Related Formulas

References

Sources

1. Dummit and Foote, Abstract Algebra
2. Herstein, Topics in Algebra
3. Wikipedia: Orbit-stabilizer theorem
4. Dummit, David S., and Richard M. Foote. Abstract Algebra. 3rd ed. John Wiley & Sons, 2004.
5. Gallian, Joseph A. Contemporary Abstract Algebra. 9th ed. Cengage Learning, 2017.
6. Dummit and Foote Abstract Algebra
7. Gallian Contemporary Abstract Algebra
8. Dummit, D. S., & Foote, R. M. (2004). Abstract Algebra (3rd ed.). John Wiley & Sons.