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nLab > Latest Changes: class equation

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- CommentRowNumber1.
- CommentAuthorTodd_Trimble
- CommentTimeJun 24th 2015
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Author: Todd_Trimble
Format: MarkdownItexNew article [[class equation]], just to fill some gaps in the nLab literature. Truly elementary stuff.

New article class equation, just to fill some gaps in the nLab literature. Truly elementary stuff.
- CommentRowNumber2.
- CommentAuthorUrs
- CommentTimeJun 24th 2015
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Author: Urs
Format: MarkdownItexNice! I have added the following remark: > Notice that reading the class equation equivalently as > $$\sum_{orbits x} \frac{{1}}{{|Stab(a_x)|}} = \frac{|A|}{|G|} $$ > it expresses the [[groupoid cardinality]] of the [[action groupoid]] of $G$ acting on $A$.

Nice!

I have added the following remark:

Notice that reading the class equation equivalently as
$<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><munder><mo lspace="0.16667em" rspace="0.16667em">∑</mo> <mrow><mi>orbits</mi><mi>x</mi></mrow></munder><mfrac><mn>1</mn><mrow><mo stretchy="false">|</mo><mi>Stab</mi><mo stretchy="false">(</mo><msub><mi>a</mi> <mi>x</mi></msub><mo stretchy="false">)</mo><mo stretchy="false">|</mo></mrow></mfrac><mo>=</mo><mfrac><mrow><mo stretchy="false">|</mo><mi>A</mi><mo stretchy="false">|</mo></mrow><mrow><mo stretchy="false">|</mo><mi>G</mi><mo stretchy="false">|</mo></mrow></mfrac></mrow><annotation encoding="application/x-tex">\sum_{orbits x} \frac{{1}}{{|Stab(a_x)|}} = \frac{|A|}{|G|} </annotation></semantics></math>$
it expresses the groupoid cardinality of the action groupoid of $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>G</mi></mrow><annotation encoding="application/x-tex">G</annotation></semantics></math>$ acting on $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>A</mi></mrow><annotation encoding="application/x-tex">A</annotation></semantics></math>$ .
- CommentRowNumber3.
- CommentAuthorTodd_Trimble
- CommentTimeJun 24th 2015
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Author: Todd_Trimble
Format: MarkdownItexThanks for the addition!

Thanks for the addition!
- CommentRowNumber4.
- CommentAuthorTodd_Trimble
- CommentTimeSep 23rd 2018
- (edited Sep 23rd 2018)
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Author: Todd_Trimble
Format: MarkdownItexReplaced a proof that I had earlier adapted from Wikipedia with a cleaner proof that I learned from Benjamin Steinberg (Theorem 3.4). For archival purposes I will post the earlier proof here. > +-- {: .proof} ###### Proof (We adapt the proof from [Wikipedia](https://en.wikipedia.org/wiki/Sylow_theorems#Proof_of_the_Sylow_theorems).) Let $P_k = \binom{G}{p^k}$ be the collection of subsets $S$ of $G$ of cardinality $p^k$, and let $G$ act on $P_k$ by taking images of left translations, $S \mapsto g S$. For $S \in P_k$, any $h \in S$ yields a monomorphism $$Stab(S) \hookrightarrow G \stackrel{g \mapsto g h}{\to} G$$ that (by definition of $Stab(S)$) factors through $S \hookrightarrow G$; this gives a monomorphism $Stab(S) \to S$, and so ${|Stab(S)|} \leq p^k$. Now we establish the reverse inequality for a suitable $S$. Writing $n = p^k m$, we have $${|P_k|} = \binom{p^k m}{p^k} = m \prod_{j = 1}^{p^k - 1} \frac{p^k m - j}{p^k - j}$$ where the product after $m$ on the right is easily seen to be prime to $p$ (any power of $p$ that divides one of the numerators $p^k m - j$ also divides the denominator $p^k - j$, so that powers of $p$ in the product are canceled). Therefore $ord_p({|P_k|}) = ord_p(m)$; let $r$ be this number. Writing out the class equation $${|P_k|} = \sum_{orbits x} \frac{{|G|}}{{|Stab(S_x)|}},$$ not every term on the right can have $p$-order greater than $r$, so there is at least one orbit $x$ where $$ord_p(\frac{{|G|}}{{|Stab(S_x)|}}) = ord_p({|G|}) - ord_p({|Stab(S_x)|}) \leq r = ord_p(m).$$ We may rearrange this inequality to say $ord_p({|G|}/m) \leq ord_p({|Stab(S_x)|})$; in other words $p^k = {|G|}/m$ divides ${|Stab(S_x)|}$. Therefore $Stab(S_x)$ has order $p^k$, which is what we wanted. =-- <a href="https://ncatlab.org/nlab/revision/diff/class+equation/10">diff</a>, <a href="https://ncatlab.org/nlab/revision/class+equation/10">v10</a>, <a href="https://ncatlab.org/nlab/show/class+equation">current</a>

Replaced a proof that I had earlier adapted from Wikipedia with a cleaner proof that I learned from Benjamin Steinberg (Theorem 3.4). For archival purposes I will post the earlier proof here.

+– {: .proof}

Proof

(We adapt the proof from Wikipedia.) Let $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>P</mi> <mi>k</mi></msub><mo>=</mo><mrow><mo>(</mo><mfrac linethickness="0px"><mi>G</mi><mrow><msup><mi>p</mi> <mi>k</mi></msup></mrow></mfrac><mo>)</mo></mrow></mrow><annotation encoding="application/x-tex">P_k = \binom{G}{p^k}</annotation></semantics></math>$ be the collection of subsets $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>S</mi></mrow><annotation encoding="application/x-tex">S</annotation></semantics></math>$ of $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>G</mi></mrow><annotation encoding="application/x-tex">G</annotation></semantics></math>$ of cardinality $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msup><mi>p</mi> <mi>k</mi></msup></mrow><annotation encoding="application/x-tex">p^k</annotation></semantics></math>$ , and let $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>G</mi></mrow><annotation encoding="application/x-tex">G</annotation></semantics></math>$ act on $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>P</mi> <mi>k</mi></msub></mrow><annotation encoding="application/x-tex">P_k</annotation></semantics></math>$ by taking images of left translations, $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>S</mi><mo>↦</mo><mi>g</mi><mi>S</mi></mrow><annotation encoding="application/x-tex">S \mapsto g S</annotation></semantics></math>$ . For $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>S</mi><mo>∈</mo><msub><mi>P</mi> <mi>k</mi></msub></mrow><annotation encoding="application/x-tex">S \in P_k</annotation></semantics></math>$ , any $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>h</mi><mo>∈</mo><mi>S</mi></mrow><annotation encoding="application/x-tex">h \in S</annotation></semantics></math>$ yields a monomorphism
$<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mi>Stab</mi><mo stretchy="false">(</mo><mi>S</mi><mo stretchy="false">)</mo><mo>↪</mo><mi>G</mi><mover><mo>→</mo><mrow><mi>g</mi><mo>↦</mo><mi>g</mi><mi>h</mi></mrow></mover><mi>G</mi></mrow><annotation encoding="application/x-tex">Stab(S) \hookrightarrow G \stackrel{g \mapsto g h}{\to} G</annotation></semantics></math>$
that (by definition of $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>Stab</mi><mo stretchy="false">(</mo><mi>S</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Stab(S)</annotation></semantics></math>$ ) factors through $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>S</mi><mo>↪</mo><mi>G</mi></mrow><annotation encoding="application/x-tex">S \hookrightarrow G</annotation></semantics></math>$ ; this gives a monomorphism $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>Stab</mi><mo stretchy="false">(</mo><mi>S</mi><mo stretchy="false">)</mo><mo>→</mo><mi>S</mi></mrow><annotation encoding="application/x-tex">Stab(S) \to S</annotation></semantics></math>$ , and so $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mrow><mo stretchy="false">|</mo><mi>Stab</mi><mo stretchy="false">(</mo><mi>S</mi><mo stretchy="false">)</mo><mo stretchy="false">|</mo></mrow><mo>≤</mo><msup><mi>p</mi> <mi>k</mi></msup></mrow><annotation encoding="application/x-tex">{|Stab(S)|} \leq p^k</annotation></semantics></math>$ . Now we establish the reverse inequality for a suitable $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>S</mi></mrow><annotation encoding="application/x-tex">S</annotation></semantics></math>$ . Writing $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>n</mi><mo>=</mo><msup><mi>p</mi> <mi>k</mi></msup><mi>m</mi></mrow><annotation encoding="application/x-tex">n = p^k m</annotation></semantics></math>$ , we have
$<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mrow><mo stretchy="false">|</mo><msub><mi>P</mi> <mi>k</mi></msub><mo stretchy="false">|</mo></mrow><mo>=</mo><mrow><mo>(</mo><mfrac linethickness="0px"><mrow><msup><mi>p</mi> <mi>k</mi></msup><mi>m</mi></mrow><mrow><msup><mi>p</mi> <mi>k</mi></msup></mrow></mfrac><mo>)</mo></mrow><mo>=</mo><mi>m</mi><munderover><mo lspace="0.16667em" rspace="0.16667em">∏</mo> <mrow><mi>j</mi><mo>=</mo><mn>1</mn></mrow> <mrow><msup><mi>p</mi> <mi>k</mi></msup><mo>−</mo><mn>1</mn></mrow></munderover><mfrac><mrow><msup><mi>p</mi> <mi>k</mi></msup><mi>m</mi><mo>−</mo><mi>j</mi></mrow><mrow><msup><mi>p</mi> <mi>k</mi></msup><mo>−</mo><mi>j</mi></mrow></mfrac></mrow><annotation encoding="application/x-tex">{|P_k|} = \binom{p^k m}{p^k} = m \prod_{j = 1}^{p^k - 1} \frac{p^k m - j}{p^k - j}</annotation></semantics></math>$
where the product after $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>m</mi></mrow><annotation encoding="application/x-tex">m</annotation></semantics></math>$ on the right is easily seen to be prime to $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>p</mi></mrow><annotation encoding="application/x-tex">p</annotation></semantics></math>$ (any power of $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>p</mi></mrow><annotation encoding="application/x-tex">p</annotation></semantics></math>$ that divides one of the numerators $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msup><mi>p</mi> <mi>k</mi></msup><mi>m</mi><mo>−</mo><mi>j</mi></mrow><annotation encoding="application/x-tex">p^k m - j</annotation></semantics></math>$ also divides the denominator $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msup><mi>p</mi> <mi>k</mi></msup><mo>−</mo><mi>j</mi></mrow><annotation encoding="application/x-tex">p^k - j</annotation></semantics></math>$ , so that powers of $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>p</mi></mrow><annotation encoding="application/x-tex">p</annotation></semantics></math>$ in the product are canceled). Therefore $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>ord</mi> <mi>p</mi></msub><mo stretchy="false">(</mo><mrow><mo stretchy="false">|</mo><msub><mi>P</mi> <mi>k</mi></msub><mo stretchy="false">|</mo></mrow><mo stretchy="false">)</mo><mo>=</mo><msub><mi>ord</mi> <mi>p</mi></msub><mo stretchy="false">(</mo><mi>m</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">ord_p({|P_k|}) = ord_p(m)</annotation></semantics></math>$ ; let $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>r</mi></mrow><annotation encoding="application/x-tex">r</annotation></semantics></math>$ be this number. Writing out the class equation
$<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mrow><mo stretchy="false">|</mo><msub><mi>P</mi> <mi>k</mi></msub><mo stretchy="false">|</mo></mrow><mo>=</mo><munder><mo lspace="0.16667em" rspace="0.16667em">∑</mo> <mrow><mi>orbits</mi><mi>x</mi></mrow></munder><mfrac><mrow><mo stretchy="false">|</mo><mi>G</mi><mo stretchy="false">|</mo></mrow><mrow><mo stretchy="false">|</mo><mi>Stab</mi><mo stretchy="false">(</mo><msub><mi>S</mi> <mi>x</mi></msub><mo stretchy="false">)</mo><mo stretchy="false">|</mo></mrow></mfrac><mo>,</mo></mrow><annotation encoding="application/x-tex">{|P_k|} = \sum_{orbits x} \frac{{|G|}}{{|Stab(S_x)|}},</annotation></semantics></math>$
not every term on the right can have $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>p</mi></mrow><annotation encoding="application/x-tex">p</annotation></semantics></math>$ -order greater than $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>r</mi></mrow><annotation encoding="application/x-tex">r</annotation></semantics></math>$ , so there is at least one orbit $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>x</mi></mrow><annotation encoding="application/x-tex">x</annotation></semantics></math>$ where
$<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><msub><mi>ord</mi> <mi>p</mi></msub><mo stretchy="false">(</mo><mfrac><mrow><mo stretchy="false">|</mo><mi>G</mi><mo stretchy="false">|</mo></mrow><mrow><mo stretchy="false">|</mo><mi>Stab</mi><mo stretchy="false">(</mo><msub><mi>S</mi> <mi>x</mi></msub><mo stretchy="false">)</mo><mo stretchy="false">|</mo></mrow></mfrac><mo stretchy="false">)</mo><mo>=</mo><msub><mi>ord</mi> <mi>p</mi></msub><mo stretchy="false">(</mo><mrow><mo stretchy="false">|</mo><mi>G</mi><mo stretchy="false">|</mo></mrow><mo stretchy="false">)</mo><mo>−</mo><msub><mi>ord</mi> <mi>p</mi></msub><mo stretchy="false">(</mo><mrow><mo stretchy="false">|</mo><mi>Stab</mi><mo stretchy="false">(</mo><msub><mi>S</mi> <mi>x</mi></msub><mo stretchy="false">)</mo><mo stretchy="false">|</mo></mrow><mo stretchy="false">)</mo><mo>≤</mo><mi>r</mi><mo>=</mo><msub><mi>ord</mi> <mi>p</mi></msub><mo stretchy="false">(</mo><mi>m</mi><mo stretchy="false">)</mo><mo>.</mo></mrow><annotation encoding="application/x-tex">ord_p(\frac{{|G|}}{{|Stab(S_x)|}}) = ord_p({|G|}) - ord_p({|Stab(S_x)|}) \leq r = ord_p(m).</annotation></semantics></math>$
We may rearrange this inequality to say $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>ord</mi> <mi>p</mi></msub><mo stretchy="false">(</mo><mrow><mo stretchy="false">|</mo><mi>G</mi><mo stretchy="false">|</mo></mrow><mo stretchy="false">/</mo><mi>m</mi><mo stretchy="false">)</mo><mo>≤</mo><msub><mi>ord</mi> <mi>p</mi></msub><mo stretchy="false">(</mo><mrow><mo stretchy="false">|</mo><mi>Stab</mi><mo stretchy="false">(</mo><msub><mi>S</mi> <mi>x</mi></msub><mo stretchy="false">)</mo><mo stretchy="false">|</mo></mrow><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">ord_p({|G|}/m) \leq ord_p({|Stab(S_x)|})</annotation></semantics></math>$ ; in other words $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msup><mi>p</mi> <mi>k</mi></msup><mo>=</mo><mrow><mo stretchy="false">|</mo><mi>G</mi><mo stretchy="false">|</mo></mrow><mo stretchy="false">/</mo><mi>m</mi></mrow><annotation encoding="application/x-tex">p^k = {|G|}/m</annotation></semantics></math>$ divides $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mrow><mo stretchy="false">|</mo><mi>Stab</mi><mo stretchy="false">(</mo><msub><mi>S</mi> <mi>x</mi></msub><mo stretchy="false">)</mo><mo stretchy="false">|</mo></mrow></mrow><annotation encoding="application/x-tex">{|Stab(S_x)|}</annotation></semantics></math>$ . Therefore $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>Stab</mi><mo stretchy="false">(</mo><msub><mi>S</mi> <mi>x</mi></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Stab(S_x)</annotation></semantics></math>$ has order $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msup><mi>p</mi> <mi>k</mi></msup></mrow><annotation encoding="application/x-tex">p^k</annotation></semantics></math>$ , which is what we wanted. =–

diff, v10, current
- CommentRowNumber5.
- CommentAuthorTodd_Trimble
- CommentTimeJan 2nd 2023
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Author: Todd_Trimble
Format: MarkdownItexAdded a corollary and a sample application of the Sylow theorems. <a href="https://ncatlab.org/nlab/revision/diff/class+equation/18">diff</a>, <a href="https://ncatlab.org/nlab/revision/class+equation/18">v18</a>, <a href="https://ncatlab.org/nlab/show/class+equation">current</a>

Added a corollary and a sample application of the Sylow theorems.

diff, v18, current
- CommentRowNumber6.
- CommentAuthorJohn Baez
- CommentTimeDec 20th 2024
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Author: John Baez
Format: MarkdownItexPointed out that the 'class formula' is also called 'Burnside's lemma', and added a link to the Wikipedia article on that. <a href="https://ncatlab.org/nlab/revision/diff/class+equation/20">diff</a>, <a href="https://ncatlab.org/nlab/revision/class+equation/20">v20</a>, <a href="https://ncatlab.org/nlab/show/class+equation">current</a>

Pointed out that the ’class formula’ is also called ’Burnside’s lemma’, and added a link to the Wikipedia article on that.

diff, v20, current

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nLab > Latest Changes: class equation

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