View source for Stabilizer

''of an element <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087090/s0870901.png" /> in a set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087090/s0870902.png" />''

The subgroup <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087090/s0870903.png" /> of a group of transformations <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087090/s0870904.png" />, operating on a set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087090/s0870905.png" />, consisting of the transformations that leave the element <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087090/s0870906.png" /> fixed: <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087090/s0870907.png" />. The stabilizer of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087090/s0870908.png" /> is also called the isotropy group of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087090/s0870909.png" />, the isotropy subgroup of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087090/s08709010.png" /> or the stationary subgroup of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087090/s08709011.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087090/s08709012.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087090/s08709013.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087090/s08709014.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087090/s08709015.png" />. If one considers the action of the group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087090/s08709016.png" /> on itself by conjugation, the stabilizer of the element <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087090/s08709017.png" /> will be the [[Centralizer|centralizer]] of this element in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087090/s08709018.png" />; if the group acts by conjugation on the set of its subgroups, then the stabilizer of a subgroup <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087090/s08709019.png" /> will be the normalizer of this subgroup (cf. [[Normalizer of a subset|Normalizer of a subset]]).



====Comments====
In case <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087090/s08709020.png" /> is a set of mathematical structures, for instance a set of lattices in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087090/s08709021.png" />, on which a group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087090/s08709022.png" /> acts, for instance the group of Euclidean motions, then the isotropy subgroup <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087090/s08709023.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087090/s08709024.png" /> is the symmetry group of the structure <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087090/s08709025.png" />.

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  L. Michel,   "Simple mathematical models for symmetry breaking"  K. Maurin (ed.)  R. Raczka (ed.) , ''Mathematical Physics and Physical Mathematics'' , Reidel  (1976)  pp. 251–262</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  S. Helgason,   "Differential geometry, Lie groups, and symmetric spaces" , Acad. Press  (1978)  pp. 121</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  T. Petrie,   J.D. Randall,   "Transformation groups on manifolds" , M. Dekker  (1984)  pp. 8, 9</TD></TR></table>