# Difference between revisions of "Orbit-stabilizer theorem"

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<cmath> \lvert G \rvert = \lvert \text{orb}(i) \rvert \cdot \lvert \text{stab}(i) \rvert . </cmath> | <cmath> \lvert G \rvert = \lvert \text{orb}(i) \rvert \cdot \lvert \text{stab}(i) \rvert . </cmath> | ||

− | ''Proof.'' Without loss of generality, let <math>G</math> operate on <math>S</math> from the | + | ''Proof.'' Without loss of generality, let <math>G</math> operate on <math>S</math> from the left. We note that if <math>\alpha, \beta</math> are elements of <math>G</math> such that <math>\alpha(i) = \beta(i)</math>, then <math>\alpha^{-1} \beta \in \text{stab}(i)</math>. Hence for any <math>x \in \text{orb}(i)</math>, the set of elements <math>\alpha</math> of <math>G</math> for which <math>\alpha(i)= x</math> constitute a unique [[coset |left coset]] modulo <math>\text{stab}(i)</math>. Thus |

<cmath> \lvert \text{orb}(i) \rvert = \lvert G/\text{stab}(i) \rvert. </cmath> | <cmath> \lvert \text{orb}(i) \rvert = \lvert G/\text{stab}(i) \rvert. </cmath> | ||

The result then follows from [[Lagrange's Theorem]]. <math>\blacksquare</math> | The result then follows from [[Lagrange's Theorem]]. <math>\blacksquare</math> | ||

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[[Category:Group theory]] | [[Category:Group theory]] | ||

+ | [[Category: Theorems]] |

## Latest revision as of 11:26, 9 April 2019

The **orbit-stabilizer theorem** is a combinatorial result in group theory.

Let be a group acting on a set . For any , let denote the stabilizer of , and let denote the orbit of . The orbit-stabilizer theorem states that

*Proof.* Without loss of generality, let operate on from the left. We note that if are elements of such that , then . Hence for any , the set of elements of for which constitute a unique left coset modulo . Thus
The result then follows from Lagrange's Theorem.