Stabilizer
of an element in a set M
The subgroup G_a of a group of transformations G, operating on a set M, (cf. Group action) consisting of the transformations that leave the element a fixed: G_a = \{ g \in G : ag = a \}. The stabilizer of a is also called the isotropy group of a, the isotropy subgroup of a or the stationary subgroup of a. If b \in M is in the orbit of a, so b = af with f \in G, then G_b = f^{-1}G_af. If one considers the action of the group G on itself by conjugation, the stabilizer of the element g \in G will be the centralizer of this element in G; if the group acts by conjugation on the set of its subgroups, then the stabilizer of a subgroup H will be the normalizer of this subgroup (cf. Normalizer of a subset).
Comments
In case M is a set of mathematical structures, for instance a set of lattices in \mathbf{R}^n, on which a group G acts, for instance the group of Euclidean motions, then the isotropy subgroup G_m of m \in M is the symmetry group of the structure m \in M.
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 |
Stabilizer. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Stabilizer&oldid=38753