Difference between revisions of "Orbit-stabilizer theorem"
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The '''orbit-stabilizer theorem''' is a [[combinatorics |combinatorial]] result in [[group theory]]. | The '''orbit-stabilizer theorem''' is a [[combinatorics |combinatorial]] result in [[group theory]]. | ||
− | Let <math>G</math> be a [[group]] acting on a [[set]] <math>S</math>. For any <math>i \in S</math>, let <math>\text{stab}(i)</math> denote the [[stabilizer]] of <math>i</math>, and let <math>\text{orb}(i)</math> denote the [[orbit]] of <math>i</math>. The orbit-stabilizer theorem states that | + | Let <math>G</math> be a [[group]] [[group action|acting]] on a [[set]] <math>S</math>. For any <math>i \in S</math>, let <math>\text{stab}(i)</math> denote the [[stabilizer]] of <math>i</math>, and let <math>\text{orb}(i)</math> denote the [[orbit]] of <math>i</math>. The orbit-stabilizer theorem states that |
<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 12:26, 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.