Orbit formula

The orbit formula is a mathematical proposition from group theory . It is often briefly summarized as: "The length of the path is the index of the stabilizer."

The set of tracks

formulation

Be a group and an operation of on a set . Then there is the picture for each ${\displaystyle \ (G,\cdot )}$ ${\displaystyle \circ :G\times M\rightarrow M}$ ${\displaystyle G}$ ${\displaystyle M}$ ${\displaystyle x\in M}$

${\displaystyle G/G_{x}\rightarrow G\circ x\ ,\ g\cdot G_{x}\mapsto g\circ x}$

a well-defined bijection . Here designated

• ${\displaystyle G\circ x:=\{g\circ x\ |\ g\in G\}\subseteq M}$the train of , ${\displaystyle x}$
• ${\displaystyle G_{x}:=\{g\in G\ |\ g\circ x=x\}\leq G}$the stabilizer from and ${\displaystyle x}$
• ${\displaystyle G/G_{x}:=\{g\cdot G_{x}\ |\ g\in G\}\subseteq {\mathcal {P}}(G)}$the set of left secondary classes of the subgroup in . ${\displaystyle G_{x}}$ ${\displaystyle G}$

proof

See: Proof of the web theorem in the evidence archive

The orbit formula is deduced from the path theorem.

Orbit formula

In the case is . Here called the index of in . The orbital formula therefore applies to finite groups ${\displaystyle |G\circ x|<\infty }$ ${\displaystyle (G:G_{x})=|G\circ x|}$ ${\displaystyle \ (G:G_{x}):=|G/G_{x}|}$ ${\displaystyle G_{x}}$ ${\displaystyle G}$ ${\displaystyle G}$

${\displaystyle \ |G|=|G\circ x|\cdot |G_{x}|}$.

Examples

conjugation

Each group operates on itself through the conjugation operation . The trajectory of an element is called the conjugation class of . The stabilizer is called the centralizer of and is denoted by. The orbital formula thus yields for finite groups ${\displaystyle G}$ ${\displaystyle g\circ x:=gxg^{-1}}$ ${\displaystyle G\circ x:=\{gxg^{-1}\ |\ g\in G\}}$ ${\displaystyle x\in G}$ ${\displaystyle x}$ ${\displaystyle G_{x}:=\{g\in G\ |\ gxg^{-1}=x\}=\{g\in G\ |\ gx=xg\}}$ ${\displaystyle x}$ ${\displaystyle Z_{G}(x)}$ ${\displaystyle G}$

${\displaystyle |G|=|G\circ x|\cdot |Z_{G}(x)|}$.

Transitive operation

If the operation of a finite group is transitive , then is ${\displaystyle G}$ ${\displaystyle M}$

${\displaystyle |M|=|G\circ x|=(G:G_{x})}$.

In this case, the thickness of must be part of the group order. ${\displaystyle M}$

See also

• Group operation
• Lagrange's theorem

literature

• Kurt Meyberg: Algebra . Part 1. 2nd edition. Carl Hanser Verlag, 1980, ISBN 3-446-13079-9 , p. 67
• Rainer Schulze-Pillot: Elementary Algebra and Number Theory . ISBN 978-3-540-45379-6 , pp. 121-124

Web links

• Eric W. Weisstein : Orbit (orbit) and orbit formula . In: MathWorld (English). (English)