# Stabilizer

*of an element in a set *

The subgroup of a group of transformations , operating on a set , consisting of the transformations that leave the element fixed: . The stabilizer of is also called the isotropy group of , the isotropy subgroup of or the stationary subgroup of . If , and , then . If one considers the action of the group on itself by conjugation, the stabilizer of the element will be the centralizer of this element in ; if the group acts by conjugation on the set of its subgroups, then the stabilizer of a subgroup will be the normalizer of this subgroup (cf. Normalizer of a subset).

#### Comments

In case is a set of mathematical structures, for instance a set of lattices in , on which a group acts, for instance the group of Euclidean motions, then the isotropy subgroup of is the symmetry group of the structure .

#### References

[a1] | L. Michel, "Simple mathematical models for symmetry breaking" K. Maurin (ed.) R. Raczka (ed.) , Mathematical Physics and Physical Mathematics , Reidel (1976) pp. 251–262 |

[a2] | S. Helgason, "Differential geometry, Lie groups, and symmetric spaces" , Acad. Press (1978) pp. 121 |

[a3] | T. Petrie, J.D. Randall, "Transformation groups on manifolds" , M. Dekker (1984) pp. 8, 9 |

**How to Cite This Entry:**

Stabilizer.

*Encyclopedia of Mathematics.*URL: http://encyclopediaofmath.org/index.php?title=Stabilizer&oldid=19013