Stabilizer (original) (raw)
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Download Wolfram NotebookLet 
be a permutation group on a set 
and 
be an element of 
. Then
 |
(1) |
|---|
is called the stabilizer of 
and consists of all the permutations of 
that produce group fixed points in 
, i.e., that send 
to itself. For example, the stabilizer of 1 and of 2 under the permutation group 
is both 
, and the stabilizer of 3 and of 4 is 
.
More generally, the subset of all images of 
under permutations of the group 
 |
(2) |
|---|
is called the group orbit of 
in 
.
A group's action on an group orbit through 
is transitive, and so is related to its isotropy group. In particular, the cosets of the isotropy subgroup correspond to the elements in the orbit,
 |
(3) |
|---|
where 
is the orbit of 
in 
and 
is the stabilizer of 
in 
. This immediately gives the identity
| 
| (4) |
| ------------------------------------------------------------------------------------------------------ | --- |
where 
denotes the order of group 
(Holton and Sheehan 1993, p. 27).
See also
Group Action, Group Fixed Point, Group Orbit, Permutation Group
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References
Holton, D. A. and Sheehan, J. Ch. 6 in The Petersen Graph. Cambridge, England: Cambridge University Press, p. 26, 1993.
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Cite this as:
Weisstein, Eric W. "Stabilizer." FromMathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/Stabilizer.html