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''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" />''
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''of an element $a$ in a set $M$''
  
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]]).
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The subgroup $G_a$ of a group of transformations $G$, operating on a set $M$, 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====
 
====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" />.
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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====
 
====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>
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<table>
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<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>
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<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>
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<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>
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</table>
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Revision as of 16:35, 1 May 2016

of an element $a$ in a set $M$

The subgroup $G_a$ of a group of transformations $G$, operating on a set $M$, 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
How to Cite This Entry:
Stabilizer. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Stabilizer&oldid=38752
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