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| '''Burnside's lemma''', sometimes also called '''Burnside's counting theorem''', the '''Cauchy-Frobenius lemma''' or the '''orbit-counting theorem''', is a result in [[group theory]] which is often useful in taking account of [[symmetry]] when counting mathematical objects. Its various eponyms include [[William Burnside]], [[George Pólya]], [[Augustin Louis Cauchy]], and [[Ferdinand Georg Frobenius]]. The result is not due to Burnside himself, who merely quotes it in his book 'On the Theory of Groups of Finite Order', attributing it instead to {{harvtxt|Frobenius|1887}}.<ref>{{harvnb|Burnside|1897|loc=§119}}</ref>
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| In the following, let ''G'' be a [[finite set|finite]] [[group (mathematics)|group]] that [[Group action|acts]] on a [[Set (mathematics)|set]] ''X''. For each ''g'' in ''G'' let ''X<sup>g</sup>'' denote the set of [[element (mathematics)|elements]] in ''X'' that are [[fixed point (mathematics)|fixed by]] ''g''. Burnside's lemma asserts the following formula for the number of [[orbit (group theory)|orbit]]s, denoted |''X''/''G''|:<ref>{{harvnb|Rotman|1995|loc=Chapter 3}}</ref>
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| :<math>|X/G| = \frac{1}{|G|}\sum_{g \in G}|X^g|.</math>
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| Thus the number of orbits (a [[natural number]] or [[Extended real number line|+∞]]) is equal to the [[mean|average]] number of points [[fixed point (mathematics)|fixed]] by an element of ''G'' (which is also a natural number or infinity). If ''G'' is infinite, the division by |''G''| may not be well-defined; in this case the following statement in [[cardinal arithmetic]] holds:
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| :<math>|G| |X/G| = \sum_{g \in G}|X^g|.</math> | |
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| == Example application ==
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| The number of rotationally distinct colourings of the faces of a [[Cube (geometry)|cube]] using three colours can be determined from this formula as follows.
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| Let ''X'' be the set of 3<sup>6</sup> possible face colour combinations that can be applied to a cube in one particular orientation, and let the rotation group ''G'' of the cube act on ''X'' in the natural manner. Then two elements of ''X'' belong to the same orbit precisely when one is simply a rotation of the other. The number of rotationally distinct colourings is thus the same as the number of orbits and can be found by counting the sizes of the [[fixed set]]s for the 24 elements of ''G''.
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| [[Image:Face colored cube.png|thumb|Cube with coloured faces]]
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| * one identity element which leaves all 3<sup>6</sup> elements of ''X'' unchanged
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| * six 90-degree face rotations, each of which leaves 3<sup>3</sup> of the elements of ''X'' unchanged
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| * three 180-degree face rotations, each of which leaves 3<sup>4</sup> of the elements of ''X'' unchanged
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| * eight 120-degree vertex rotations, each of which leaves 3<sup>2</sup> of the elements of ''X'' unchanged
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| * six 180-degree edge rotations, each of which leaves 3<sup>3</sup> of the elements of ''X'' unchanged
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| A detailed examination of these automorphisms may be found
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| [[Cycle_index#The_cycle_index_of_the_face_permutations_of_a_cube|here]].
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| The average fix size is thus
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| : <math> \frac{1}{24}\left(3^6+6\cdot 3^3 + 3 \cdot 3^4 + 8 \cdot 3^2 + 6 \cdot 3^3 \right) = 57. </math>
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| Hence there are 57 rotationally distinct colourings of the faces of a cube in three colours. In general, the number of rotationally distinct colorings of the faces of a cube in ''n'' colors is given by
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| : <math> \frac{1}{24}\left(n^6+3n^4 + 12n^3 + 8n^2\right). </math>
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| == Proof ==
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| The proof uses the [[orbit-stabilizer theorem]] and the fact that ''X'' is the disjoint union of the orbits:
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| <math>\begin{align}
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| \sum_{g \in G}|X^g| &= |\{(g,x)\in G\times X \mid g\cdot x = x\}| = \sum_{x \in X} |G_x| = \sum_{x \in X} \frac{|G|}{|Gx|} \\
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| &= |G| \sum_{x \in X}\frac{1}{|Gx|} = |G|\sum_{A\in X/G}\sum_{x\in A} \frac{1}{|A|} = |G| \sum_{A\in X/G} 1 \\
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| &= |G| \cdot |X/G|.
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| \end{align}</math>
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| ==History: the lemma that is not Burnside's==
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| [[William Burnside]] stated and proved this lemma, attributing it to {{harvnb|Frobenius|1887}} in his 1897 book on finite groups. But even prior to Frobenius, the formula was known to [[Cauchy]] in 1845. In fact, the lemma was apparently so well known that Burnside simply omitted to attribute it to Cauchy. Consequently, this lemma is sometimes referred to as '''[[Stigler's law of eponymy|the lemma that is not Burnside's]]'''.<ref>{{harvnb|Neumann|1979}}</ref> This is less ambiguous than it may seem: Burnside contributed many lemmas to this field.
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| == See also ==
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| * [[Pólya enumeration theorem]]
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| ==Notes==
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| {{reflist}}
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| == References ==
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| * {{Citation | last1=Burnside | first1=William | author1-link=William Burnside | title=Theory of groups of finite order | publisher=[[Cambridge University Press]] | year=1897}}. (This is the first edition; the introduction to the second edition contains Burnside's famous ''volte face'' regarding the utility of [[representation theory]].)
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| * {{citation | last=Frobenius |first=Ferdinand Georg |authorlink=Ferdinand Georg Frobenius |title=Ueber die Congruenz nach einem aus zwei endlichen Gruppen gebildeten Doppelmodul |journal=Crelle |volume=CI |year=1887 |page=288}}.
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| * {{Citation | last1=Neumann | first1=Peter M. | author1-link=Peter M. Neumann | title=A lemma that is not Burnside's | mr=562002 | year=1979 | journal=The Mathematical Scientist | issn=0312-3685 | volume=4 | issue=2 | pages=133–141}}.
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| * {{citation | last=Rotman |first=Joseph |title=An introduction to the theory of groups |publisher=Springer-Verlag |year=1995 |isbn=0-387-94285-8}}.
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| [[Category:Lemmas]]
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| [[Category:Group theory]]
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