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| In [[mathematics]], the '''Burnside ring''' of a [[finite group]] is an algebraic construction that encodes the different ways the group can [[group action|act]] on finite sets. The ideas were introduced by [[William Burnside]] at the end of the nineteenth century, but the algebraic [[Ring (mathematics)|ring structure]] is a more recent development, due to Solomon (1967).
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| ==Formal definition==
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| Given a [[finite group]] ''G'', the elements of its Burnside ring ''Ω''(''G'') are the formal differences of isomorphism classes of finite [[group action|''G''-sets]]. For the [[Ring (mathematics)|ring structure]], addition is given by [[disjoint union]] of ''G''-sets, and multiplication by their [[Cartesian product]].
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| The Burnside ring is a free '''Z'''-[[Module (mathematics)|module]], whose generators are the (isomorphism classes of) [[Group action|orbit types]] of ''G''.
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| If ''G'' acts on a finite set ''X'', then one can write <math>X = \cup_i X_i</math> (disjoint union), where each ''X''<sub>''i''</sub> is a single ''G''-orbit. Choosing any element ''x''<sub>''i''</sub> in ''X''<sub>i</sub> creates an isomorphism ''G''/''G''<sub>''i''</sub> → ''X''<sub>''i''</sub>, where ''G<sub>i</sub>'' is the stabilizer (isotropy) subgroup of ''G'' at ''x''<sub>''i''</sub>. A different choice of representative ''y''<sub>''i''</sub> in ''X''<sub>''i''</sub> gives a conjugate subgroup to ''G''<sub>''i''</sub> as stabilizer. This shows that the generators of ''Ω(G)'' as a '''Z'''-module are the orbits ''G''/''H'' as ''H'' ranges over [[conjugacy classes]] of subgroups of ''G''.
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| In other words, a typical element of ''Ω''(''G'') is
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| :<math> \sum_{i=1}^N a_i [G/G_i],</math>
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| where ''a''<sub>''i''</sub> in '''Z''' and ''G''<sub>1</sub>, ''G''<sub>2</sub>, ..., ''G''<sub>''N''</sub> are representatives of the conjugacy classes of subgroups of ''G''.
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| ==Marks==
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| Much like [[character theory]] simplifies working with [[group representation]]s, '''marks''' simplify working with [[permutation representation]]s and the Burnside ring.
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| If ''G'' acts on ''X'', and ''H'' ≤ ''G'' (''H'' is a [[subgroup]] of ''G''), then the '''mark''' of ''H'' on ''X'' is the number of elements of ''X'' that are fixed by every element of ''H'': <math>m_X(H) = \left|X^H\right|</math>, where
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| :<math>X^H = \{ x\in X \mid h\cdot x = x, \forall h\in H\}.</math>
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| If ''H'' and ''K'' are conjugate subgroups, then ''m''<sub>''X''</sub>(''H'') = ''m''<sub>''X''</sub>(''K'') for any finite ''G''-set ''X''; indeed, if ''K'' = ''gHg''<sup>−1</sup> then ''X''<sup>''K''</sup> = ''g'' · ''X''<sup>''H''</sup>.
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| It is also easy to see that for each ''H'' ≤ ''G'', the map ''Ω''(''G'') → '''Z''' : ''X'' ↦ ''m''<sub>''X''</sub>(''H'') is a homomorphism. This means that to know the marks of ''G'', it is sufficient to evaluate them on the generators of ''Ω''(''G''), ''viz.'' the orbits ''G''/''H''.
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| For each pair of subgroups ''H'',''K'' ≤ ''G'' define
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| :<math>m(K, H) = \left|[G/K]^H\right| = \# \left\{ gK \in G/K \mid HgK=gK \right\}.</math>
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| This is ''m''<sub>''X''</sub>(''H'') for ''X'' = ''G''/''K''. The condition ''HgK'' = ''gK'' is equivalent to ''g''<sup>−1</sup>''Hg'' ≤ ''K'', so if ''H'' is not conjugate to a subgroup of ''K'' then ''m''(''K'', ''H'') = 0.
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| To record all possible marks, one forms a table, Burnside's '''Table of Marks''', as follows: Let ''G''<sub>1</sub> (= trivial subgroup), ''G''<sub>2</sub>, ..., ''G''<sub>''N''</sub> = ''G'' be representatives of the ''N'' conjugacy classes of subgroups of ''G'', ordered in such a way that whenever ''G''<sub>''i''</sub> is conjugate to a subgroup of ''G''<sub>''j''</sub>, then ''i'' ≤ ''j''. Now define the ''N'' × ''N'' table (square matrix) whose (''i'', ''j'')th entry is ''m''(''G''<sub>''i''</sub>, ''G''<sub>''j''</sub>). This matrix is lower triangular, and the elements on the diagonal are non-zero so it is invertible.
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| It follows that if ''X'' is a ''G''-set, and '''u''' its row vector of marks, so ''u''<sub>''i''</sub> = ''m''<sub>''X''</sub>(''G''<sub>''i''</sub>), then ''X'' decomposes as a [[disjoint union]] of ''a''<sub>''i''</sub> copies of the orbit of type ''G''<sub>''i''</sub>, where the vector '''a''' satisfies,
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| :'''a'''''M'' = '''''u''''', | |
| where ''M'' is the matrix of the table of marks. This theorem is due to {{harv|Burnside|1897}}.
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| ==Examples==
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| The table of marks for the cyclic group of order 6:
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| {| style="background:white; color:lightseagreen; width:12em; height:12em; text-align:center;"
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| |- style="background:#abc; color:white; "
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| | style="background:#345; color:white;" | '''Z'''<sub>6</sub> || '''1''' || '''Z'''<sub>2</sub> ||'''Z'''<sub>3</sub> || '''Z'''<sub>6</sub>
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| |-
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| | style="background:#abc; color:white;" | '''Z'''<sub>6</sub> / '''1''' || 6 || . ||. ||.
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| |-
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| | style="background:#abc; color:white;" | '''Z'''<sub>6</sub> / '''Z'''<sub>2</sub> || 3 || 3 ||. ||.
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| |-
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| | style="background:#abc; color:white;" | '''Z'''<sub>6</sub> / '''Z'''<sub>3</sub> || 2 || 0 || 2 || .
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| |-
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| | style="background:#abc; color:white;" | '''Z'''<sub>6</sub> / '''Z'''<sub>6</sub> || 1 || 1 || 1 || 1
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| |}
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| The table of marks for the symmetric group ''S<sub>3</sub>'' on 3 letters:
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| {| style="background:white; color:lightseagreen; width:12em; height:12em; text-align:center"
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| |- style="background:#abc; color:white;"
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| | style="background:#345; color:white;" | '''S'''<sub>3</sub> || '''1''' || '''Z'''<sub>2</sub> ||'''Z'''<sub>3</sub> || '''S'''<sub>3</sub>
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| |-
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| | style="background:#abc; color:white;" | '''S'''<sub>3</sub> / '''1''' || 6 || . ||. ||.
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| |-
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| | style="background:#abc; color:white;" | '''S'''<sub>3</sub> / '''Z'''<sub>2</sub> || 3 || 1 ||. ||.
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| |-
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| | style="background:#abc; color:white;" | '''S'''<sub>3</sub> / '''Z'''<sub>3</sub> || 2 || 0 || 2 || .
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| |-
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| | style="background:#abc; color:white;" | '''S'''<sub>3</sub> / '''S'''<sub>3</sub> || 1 || 1 || 1 || 1
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| |}
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| The dots in the two tables are all zeros, merely emphasizing the fact that the tables are lower-triangular.
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| (Some authors use the transpose of the table, but this is how Burnside defined it originally.)
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| The fact that the last row is all 1s is because [''G''/''G''] is a single point. The diagonal terms are ''m''(''H'', ''H'') = | ''N''<sub>''G''</sub>(''H'')/''H'' |.
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| The ring structure of ''Ω''(''G'') can be deduced from these tables: the generators of the ring (as a '''Z'''-module) are the rows of the table, and the product of two generators has mark given by the product of the marks (so component-wise multiplication of row vectors), which can then be decomposed as a [[linear combination]] of all the rows. For example, with ''S''<sub>3</sub>,
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| :<math>[G/\mathbf{Z}_2]\cdot[G/\mathbf{Z}_3] = [G/1],</math>
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| as (3, 1, 0, 0).(2, 0, 2, 0) = (6, 0, 0, 0).
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| ==Permutation representations==
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| Associated to any finite set ''X'' is a [[vector space]] ''V = V<sub>X</sub>'', which is the vector space with the elements of ''X'' as the basis (using any specified field). An action of a finite group ''G'' on ''X'' induces a linear action on ''V'', called a permutation [[Group representation|representation]]. The set of all finite dimensional representations of ''G'' has the structure of a ring, the [[representation ring]], denoted ''R(G)''.
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| For a given ''G''-set ''X'', the [[Character theory|character]] of the associated representation is
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| :<math>\chi(g) = m_X(\langle g\rangle)</math>
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| where <''g''> is the cyclic group generated by ''g''. | |
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| The resulting map
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| :<math>\beta : \Omega(G) \longrightarrow R(G) </math>
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| taking a ''G''-set to the corresponding representation is in general neither injective nor surjective.
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| The simplest example showing that β is not in general injective is for ''G = S<sub>3</sub>'' (see table above), and is given by
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| :<math>\beta(2[S_3/\mathbf{Z}_2] + [S_3/\mathbf{Z}_3]) = \beta([S_3] + 2[S_3/S_3]).</math>
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| ==Extensions==
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| The Burnside ring for [[compact group]]s is described in {{harv|tom Dieck|1987}}.
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| The [[Segal conjecture]] relates the Burnside ring to [[homotopy]].
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| == See also ==
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| *[[Burnside category]]
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| ==References==
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| *{{Citation | last=Burnside | first=William | author-link=William Burnside | title=Theory of groups of finite order | publisher=[[Cambridge University Press]] | year=1897}}
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| *{{Citation | last=tom Dieck | first=Tammo | title=Transformation groups | publisher=Walter de Gruyter | series=de Gruyter Studies in Mathematics | isbn=978-3-11-009745-0 | mr=889050 | year=1987 | volume=8 | oclc=217014538 | unused_data=isbn status=May be invalid - please double check}}<!-- alphabetize under "Dieck", but list as "tom Dieck" -->
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| *{{Citation | last1=Dress | first1=Andreas | title=A characterization of solvable groups | journal=Math. Zeitschrift | year=1969 | pages=213–217 | doi=10.1007/BF01110213 | volume=110 | issue=3}}
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| *{{Citation | last1=Kerber | first1=Adalbert | title=Applied finite group actions | publisher=[[Springer-Verlag]] | location=Berlin, New York | edition=2nd | series=Algorithms and Combinatorics | isbn=978-3-540-65941-9 | mr=1716962 | year=1999 | volume=19 | oclc=247593131 | unused_data=isbn status=May be invalid - please double check}}
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| *{{Citation | last1=Solomon | first1= L. | title=The Burnside algebra of a finite group | journal=J. Combin. Theory | year=1967 | pages=603–615 | volume=1}}
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| [[Category:Group theory]]
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| [[Category:Finite groups]]
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| [[Category:Permutation groups]]
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| [[Category:Representation theory of groups]]
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