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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).


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 .


[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:
This article was adapted from an original article by N.N. Vil'yams (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article