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
a well-defined bijection . Here designated
- the train of ,
- the stabilizer from and
- the set of left secondary classes of the subgroup in .
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
- .
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
- .
Transitive operation
If the operation of a finite group is transitive , then is
- .
In this case, the thickness of must be part of the group order.
See also
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)