https://artofproblemsolving.com/wiki/index.php?title=Stabilizer&feed=atom&action=historyStabilizer - Revision history2024-03-28T21:09:26ZRevision history for this page on the wikiMediaWiki 1.31.1https://artofproblemsolving.com/wiki/index.php?title=Stabilizer&diff=27874&oldid=prevJam: Link to group action article2008-09-09T22:47:46Z<p>Link to group action article</p>
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<tr><td class='diff-marker'>−</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>A '''stabilizer''' is a part of a [[monoid]] (or [[group]]) acting on a set.</div></td><td class='diff-marker'>+</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>A '''stabilizer''' is a part of a [[monoid]] (or [[group]]) <ins class="diffchange diffchange-inline">[[group action|</ins>acting<ins class="diffchange diffchange-inline">]] </ins>on a set.</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>Specifically, let <math>M</math> be a monoid operating on a set <math>S</math>, and let <math>A</math> be a subset of <math>S</math>.  The ''stabilizer'' of <math>A</math>, sometimes denoted <math>\text{stab}(A)</math>, is the set of elements of <math>a</math> of <math>M</math> for which <math>a(A) \subseteq  A</math>; the ''strict stabilizer''' is the set of <math>a \in M</math> for which <math>a(A)=A</math>.  In other words, the stabilizer of <math>A</math> is the [[transporter]] of <math>A</math> to itself.</div></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>Specifically, let <math>M</math> be a monoid operating on a set <math>S</math>, and let <math>A</math> be a subset of <math>S</math>.  The ''stabilizer'' of <math>A</math>, sometimes denoted <math>\text{stab}(A)</math>, is the set of elements of <math>a</math> of <math>M</math> for which <math>a(A) \subseteq  A</math>; the ''strict stabilizer''' is the set of <math>a \in M</math> for which <math>a(A)=A</math>.  In other words, the stabilizer of <math>A</math> is the [[transporter]] of <math>A</math> to itself.</div></td></tr>
</table>Jamhttps://artofproblemsolving.com/wiki/index.php?title=Stabilizer&diff=26106&oldid=prevBoy Soprano II: small note on inner automorphisms2008-05-22T03:07:42Z<p>small note on inner automorphisms</p>
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<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div><cmath> \text{stab}(ax) \subseteq  a\, \text{stab}(x) a^{-1} , </cmath></div></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div><cmath> \text{stab}(ax) \subseteq  a\, \text{stab}(x) a^{-1} , </cmath></div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>whence the desired result.  <math>\blacksquare</math></div></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>whence the desired result.  <math>\blacksquare</math></div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;"></ins></div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">In other words, the stabilizer of <math>ax</math> is the image of the stabilizer of <math>x</math> under the [[inner automorphism]] <math>\text{Int}(a)</math>.</ins></div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td></tr>
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</table>Boy Soprano IIhttps://artofproblemsolving.com/wiki/index.php?title=Stabilizer&diff=26104&oldid=prevBoy Soprano II: added a proposition2008-05-22T02:33:09Z<p>added a proposition</p>
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<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>Specifically, let <math>M</math> be a monoid operating on a set <math>S</math>, and let <math>A</math> be a subset of <math>S</math>.  The ''stabilizer'' of <math>A</math>, sometimes denoted <math>\text{stab}(A)</math>, is the set of elements of <math>a</math> of <math>M</math> for which <math>a(A) \subseteq  A</math>; the ''strict stabilizer''' is the set of <math>a \in M</math> for which <math>a(A)=A</math>.  In other words, the stabilizer of <math>A</math> is the [[transporter]] of <math>A</math> to itself.</div></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>Specifically, let <math>M</math> be a monoid operating on a set <math>S</math>, and let <math>A</math> be a subset of <math>S</math>.  The ''stabilizer'' of <math>A</math>, sometimes denoted <math>\text{stab}(A)</math>, is the set of elements of <math>a</math> of <math>M</math> for which <math>a(A) \subseteq  A</math>; the ''strict stabilizer''' is the set of <math>a \in M</math> for which <math>a(A)=A</math>.  In other words, the stabilizer of <math>A</math> is the [[transporter]] of <math>A</math> to itself.</div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;"></ins></div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">By abuse of language, for an element <math>x\in S</math>, the stabilizer of <math>\{x\}</math> is called the stabilizer of <math>x</math>.</ins></div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>The stabilizer of any set <math>A</math> is evidently a sub-monoid of <math>M</math>, as is the strict stabilizer.  Also, if <math>a</math> is an invertible element of <math>M</math> and a member of the strict stabilizer of <math>A</math>, then <math>a^{-1}</math> is also an element of the strict stabilizer of <math>a</math>, for the restriction of the function <math>a : S \to S</math> to <math>A</math> is a bijection from <math>A</math> to itself.</div></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>The stabilizer of any set <math>A</math> is evidently a sub-monoid of <math>M</math>, as is the strict stabilizer.  Also, if <math>a</math> is an invertible element of <math>M</math> and a member of the strict stabilizer of <math>A</math>, then <math>a^{-1}</math> is also an element of the strict stabilizer of <math>a</math>, for the restriction of the function <math>a : S \to S</math> to <math>A</math> is a bijection from <math>A</math> to itself.</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td></tr>
<tr><td class='diff-marker'>−</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>It follows that if <math>M</math> is a group <math>G</math>, then the strict stabilizer of <math>A</math> is a [[subgroup]] of <math>G</math>, since every element of <math>G</math> is a bijection on <math>S</math>, but the stabilizer need not be.  For example, let <math>G=S= \mathbb{Z}</math>, with <math>g(s) = g+s</math>, and let <math>A=\mathbb{Z}_{>0}</math>.  Then the stabilizer of <math>A</math> is the set of nonnegative <del class="diffchange diffchange-inline">integers</del>, which is evidently not a group.  On the other hand, the strict stabilizer of <math>A</math> is the set <math>\{0\}</math>, the trivial group.  On the other hand, if <math>A</math> is ''finite'', then the strict stabilizer and the stabilizer are one and the same, since <math>a : S \to S</math> is bijective, for all <math>a\in G</math>.</div></td><td class='diff-marker'>+</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>It follows that if <math>M</math> is a <ins class="diffchange diffchange-inline">[[</ins>group<ins class="diffchange diffchange-inline">]] </ins><math>G</math>, then the strict stabilizer of <math>A</math> is a [[subgroup]] of <math>G</math>, since every element of <math>G</math> is a <ins class="diffchange diffchange-inline">[[</ins>bijection<ins class="diffchange diffchange-inline">]] </ins>on <math>S</math>, but the stabilizer need not be.  For example, let <math>G=S= \mathbb{Z}</math>, with <math>g(s) = g+s</math>, and let <math>A=\mathbb{Z}_{>0}</math>.  Then the stabilizer of <math>A</math> is the set of nonnegative <ins class="diffchange diffchange-inline">[[integer]]s</ins>, which is evidently not a group.  On the other hand, the strict stabilizer of <math>A</math> is the set <math>\{0\}</math>, the trivial group.  On the other hand, if <math>A</math> is ''finite'', then the strict stabilizer and the stabilizer are one and the same, since <math>a : S \to S</math> is bijective, for all <math>a\in G</math>.</div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div> </div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins class="diffchange diffchange-inline">'''Proposition.''' Let <math>G</math> be a group acting on a set <math>S</math>.  Then for all <math>x\in S</math> and all <math>a \in G</math>, <math>\text{stab}(ax) = a\, \text{stab}(x) a^{-1}</math>.</ins></div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div> </div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins class="diffchange diffchange-inline">''Proof.''  Note that for any <math>g \in \text{stab}(x)</math>, <math>(aga^{-1})ax = agx = ax.</math>  It follows that</ins></div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins class="diffchange diffchange-inline"><cmath> a\, \text{stab}(x) a^{-1} \subseteq \text{stab}(ax) . </cmath></ins></div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins class="diffchange diffchange-inline">By simultaneously replacing <math>x</math> with <math>ax</math> and <math>a</math> with <math>a^{-1}</math>, we have</ins></div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins class="diffchange diffchange-inline"><cmath> \text{stab}(ax) \subseteq  a\, \text{stab}(x) a^{-1} , </cmath></ins></div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins class="diffchange diffchange-inline">whence the desired result.  <math>\blacksquare</math></ins></div></td></tr>
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</table>Boy Soprano IIhttps://artofproblemsolving.com/wiki/index.php?title=Stabilizer&diff=25995&oldid=prevBoy Soprano II: start2008-05-14T23:52:40Z<p>start</p>
<p><b>New page</b></p><div>A '''stabilizer''' is a part of a [[monoid]] (or [[group]]) acting on a set.<br />
<br />
Specifically, let <math>M</math> be a monoid operating on a set <math>S</math>, and let <math>A</math> be a subset of <math>S</math>. The ''stabilizer'' of <math>A</math>, sometimes denoted <math>\text{stab}(A)</math>, is the set of elements of <math>a</math> of <math>M</math> for which <math>a(A) \subseteq A</math>; the ''strict stabilizer''' is the set of <math>a \in M</math> for which <math>a(A)=A</math>. In other words, the stabilizer of <math>A</math> is the [[transporter]] of <math>A</math> to itself.<br />
<br />
The stabilizer of any set <math>A</math> is evidently a sub-monoid of <math>M</math>, as is the strict stabilizer. Also, if <math>a</math> is an invertible element of <math>M</math> and a member of the strict stabilizer of <math>A</math>, then <math>a^{-1}</math> is also an element of the strict stabilizer of <math>a</math>, for the restriction of the function <math>a : S \to S</math> to <math>A</math> is a bijection from <math>A</math> to itself.<br />
<br />
It follows that if <math>M</math> is a group <math>G</math>, then the strict stabilizer of <math>A</math> is a [[subgroup]] of <math>G</math>, since every element of <math>G</math> is a bijection on <math>S</math>, but the stabilizer need not be. For example, let <math>G=S= \mathbb{Z}</math>, with <math>g(s) = g+s</math>, and let <math>A=\mathbb{Z}_{>0}</math>. Then the stabilizer of <math>A</math> is the set of nonnegative integers, which is evidently not a group. On the other hand, the strict stabilizer of <math>A</math> is the set <math>\{0\}</math>, the trivial group. On the other hand, if <math>A</math> is ''finite'', then the strict stabilizer and the stabilizer are one and the same, since <math>a : S \to S</math> is bijective, for all <math>a\in G</math>.<br />
<br />
{{stub}}<br />
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== See also ==<br />
<br />
* [[Orbit-stabilizer theorem]]<br />
<br />
[[Category:Abstract algebra]]<br />
[[Category:Group theory]]</div>Boy Soprano II