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 |
Stabilizer. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Stabilizer&oldid=38752