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Stabilizer


Let G be a permutation group on a set Omega and x be an element of Omega. Then

 G_x={g in G:g(x)=x}
(1)

is called the stabilizer of x and consists of all the permutations of G that produce group fixed points in x, i.e., that send x to itself. For example, the stabilizer of 1 and of 2 under the permutation group {(1)(2)(3)(4),(12)(3)(4),(1)(2)(34),(12)(34)} is both {(1)(2)(3)(4),(1)(2)(34)}, and the stabilizer of 3 and of 4 is {(1)(2)(3)(4),(12)(3)(4)}.

More generally, the subset of all images of x in Omega under permutations of the group G

 G(x)={g(x):g in G}
(2)

is called the group orbit of x in G.

A group's action on an group orbit through x 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,

 G(x)∼G/G_x,
(3)

where G(x) is the orbit of x in G and G_x is the stabilizer of x in G. This immediately gives the identity

 |G|=|G_x||G(x)|,
(4)

where |G| denotes the order of group G (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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Stabilizer

Cite this as:

Weisstein, Eric W. "Stabilizer." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/Stabilizer.html

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