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| {{Group theory sidebar |Basics}}
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| In [[mathematics]], specifically in [[group theory]], the concept of a '''semidirect product''' is a generalization of a [[direct product of groups|direct product]]. There are two closely related concepts of semidirect product: an ''inner'' semidirect product is a particular way in which a [[Group (mathematics)|group]] can be constructed from two [[subgroup]]s, one of which is a [[normal subgroup]], while an ''outer'' semidirect product is a [[cartesian product]] as a set, but with a particular multiplication operation. As with direct products, there is a natural equivalence between inner and outer semidirect products, and both are commonly referred to simply as ''semidirect products''.
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| == Some equivalent definitions of inner semidirect products ==
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| Let ''G'' be a group with [[identity element]] ''e'', ''N'' a [[normal subgroup]] of ''G'' (i.e., ''N'' ◁ ''G'') and ''H'' a [[subgroup]] of ''G''. The following statements are equivalent:
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| * ''G'' = ''NH'' and ''N'' ∩ ''H'' = {''e''}.
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| * Every element of ''G'' can be written in a unique way as a product ''nh'', with ''n'' ∈ ''N'' and ''h'' ∈ ''H''.
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| * Every element of ''G'' can be written in a unique way as a product ''hn'', with ''h'' ∈ ''H'' and ''n'' ∈ ''N''.
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| * The natural embedding ''H'' → ''G'', composed with the natural projection ''G'' → ''G / N'', yields an [[group isomorphism|isomorphism]] between ''H'' and the [[quotient group]] ''G / N''.
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| * There exists a [[group homomorphism|homomorphism]] ''G'' → ''H'' which is the identity on ''H'' and whose [[kernel (algebra)|kernel]] is ''N''.
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| If one (and therefore all) of these statements hold, we say that ''G'' is a '''semidirect product''' of ''N'' and ''H'', written <math>G = N \rtimes H,</math> or that ''G'' ''splits'' over ''N'';
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| one also says that ''G'' is a '''semidirect''' product of ''H'' acting on ''N'', or even a semidirect product of ''H'' and ''N''. In order to avoid ambiguities, it is advisable to specify which of the
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| two subgroups is normal.
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| == Elementary facts and caveats ==
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| If ''G'' is the semidirect product of the normal subgroup ''N'' and the subgroup ''H'', and both ''N'' and ''H'' are finite, then the [[order of a group|order]] of ''G'' equals the product of the orders of ''N'' and ''H''.
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| Note that, as opposed to the case with the [[direct product of groups|direct product]], a semidirect product of two groups is not, in general, unique; if ''G'' and ''G′'' are two groups which both contain isomorphic copies of ''N'' as a normal subgroup and ''H'' as a subgroup, and both are a semidirect product of ''N'' and ''H'', then it does ''not'' follow that ''G'' and ''G′'' are [[group isomorphism|isomorphic]]. This remark leads to an [[extension problem]], of describing the possibilities.
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| == Semidirect products and group homomorphisms ==
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| Let ''G'' be a semidirect product of the normal subgroup ''N'' and the subgroup ''H''. Let Aut(''N'') denote the group of all [[automorphism]]s of ''N''. The map φ : ''H'' → Aut(''N'') defined by φ(''h'') = φ<sub>''h''</sub>, where φ<sub>''h''</sub>(''n'') = ''hnh''<sup>−1</sup> for all ''h'' in ''H'' and ''n'' in ''N'', is a [[group homomorphism]]. (Note that ''hnh''<sup>−1</sup>∈''N'' since ''N'' is normal in ''G''.) Together ''N'', ''H'' and φ determine ''G'' [[up to]] isomorphism, as we show now.
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| Given any two groups ''N'' and ''H'' (not necessarily subgroups of a given group) and a group homomorphism <math>\varphi</math> : ''H'' → Aut(''N''), we can construct a new group <math>N\rtimes_{\varphi}H</math>, called the '''(outer) semidirect product of ''N'' and ''H'' with respect to <math>\varphi</math>''', defined as follows.<ref>{{cite book |last1=Robinson |first1=Derek John Scott |title=An Introduction to Abstract Algebra |year=2003 |publisher=Walter de Gruyter |isbn=9783110175448 |pages=75--76}}</ref>
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| *As a set, <math>N\rtimes_{\varphi}H</math> is the [[cartesian product]] ''N'' × ''H''.
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| *Multiplication of elements in <math>N\rtimes_{\varphi}H</math> is determined by the homomorphism <math>\varphi</math>. The operation is
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| ::<math>*\colon (N\rtimes_{\varphi} H)\times(N\rtimes_{\varphi} H)\to N\rtimes_{\varphi} H</math>
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| :defined by
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| ::<math>(n_1, h_1)*(n_2, h_2) = (n_1\varphi_{h_1}(n_2), h_1h_2)</math>
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| :for ''n''<sub>1</sub>, ''n''<sub>2</sub> in ''N'' and ''h''<sub>1</sub>, ''h''<sub>2</sub> in ''H''.
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| This defines a group in which the identity element is (''e''<sub>''N''</sub>, ''e''<sub>''H''</sub>) and the inverse of the element (''n'', ''h'') is (<math>\varphi</math><sub>''h''<sup>–1</sup></sub>(''n''<sup>–1</sup>), ''h''<sup>–1</sup>). Pairs (''n'',''e''<sub>''H''</sub>) form a normal subgroup isomorphic to ''N'', while pairs (''e''<sub>''N''</sub>, ''h'') form a subgroup isomorphic to ''H''. The full group is a semidirect product of those two subgroups in the sense given earlier.
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| Conversely, suppose that we are given a group ''G'' with a normal subgroup ''N'' and a subgroup ''H'', such that every element ''g'' of ''G'' may be written uniquely in the form ''g=nh'' where ''n'' lies in ''N'' and ''h'' lies in ''H''. Let <math>\varphi</math>: ''H'' → Aut(''N'') be the homomorphism given by <math>\varphi</math>(''h'') = <math>\varphi</math><sub>''h''</sub>, where
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| :<math>\varphi_h(n) = hnh^{-1}</math>
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| for all ''n'' in ''N'' and ''h'' in ''H''.
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| Then ''G'' is isomorphic to the semidirect product <math>N\rtimes_{\varphi}H</math> ; the isomorphism sends the product ''nh'' to the tuple (''n'',''h''). In ''G'', we have
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| :<math>(n_1h_1)(n_2h_2) = n_1 h_1 n_2 h_1^{-1}h_1h_2 = (n_1\varphi_{h_1}(n_2))(h_1h_2)</math>
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| which shows that the above map is indeed an isomorphism and also explains the definition of the multiplication rule in <math>N\rtimes_{\varphi}H</math>.
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| A version of the [[splitting lemma]] for groups states that a group ''G'' is isomorphic to a semidirect product of the two groups ''N'' and ''H'' if and only if there exists a [[exact sequence|short exact sequence]]
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| :<math> 1\longrightarrow N \longrightarrow^{\!\!\!\!\!\!\!\!\!\beta}\ \, G \longrightarrow^{\!\!\!\!\!\!\!\!\!\alpha}\ \, H \longrightarrow 1</math>
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| and a group homomorphism γ : ''H'' → ''G'' such that <math>\alpha \circ \gamma = \operatorname{id}_H</math>, the [[identity function|identity map]] on ''H''. In this case, <math>\varphi</math> : ''H'' → Aut(''N'') is given by <math>\varphi</math>(''h'') = <math>\varphi</math><sub>''h''</sub>, where
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| :<math>\varphi_h(n) = \beta^{-1}(\gamma(h)\beta(n)\gamma(h^{-1})).</math>
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| If <math>\varphi</math> is the trivial homomorphism, sending every element of ''H'' to the identity automorphism of ''N'', then <math>N\rtimes_{\varphi}H</math> is the direct product <math> N \times H</math>.
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| == Examples ==
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| The [[dihedral group]] ''D''<sub>2''n''</sub> with 2''n'' elements is isomorphic to a semidirect product of the [[cyclic group]]s ''C''<sub>''n''</sub> and ''C''<sub>2</sub>.<ref name="mac-lane">{{cite book |last1=Mac Lane |first1=Saunders |authorlink1=Saunders Mac Lane |last2=Birkhoff |first2=Garrett |authorlink2=Garrett Birkhoff |title=Algebra |edition=3 |year=1999 |publisher=American Mathematical Society |isbn=0-8218-1646-2 |pages=414--415}}</ref> Here, the non-identity element of ''C''<sub>2</sub> acts on ''C''<sub>''n''</sub> by inverting elements; this is an automorphism since ''C''<sub>''n''</sub> is [[abelian group|abelian]]. The [[group presentation|presentation]] for this group is:
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| :<math>\langle a,\;b \mid a^2 = e,\; b^n = e,\; aba^{-1}=b^{-1}\rangle.</math>
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| More generally, a semidirect product of any two cyclic groups <math>C_m\;</math> with generator <math>a\;</math> and <math>C_n\;</math> with generator <math>b\;</math> is given by a single relation <math>aba^{-1}=b^k\;</math> with <math>k\;</math> and <math>n\;</math> [[coprime]], i.e. the presentation:<ref name="mac-lane" />
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| :<math>\langle a,\;b \mid a^m=e,\;b^n = e,\;aba^{-1}=b^k\;\rangle.</math>
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| If <math>r\;</math> and <math>m\;</math> are coprime, <math>a^r\;</math> is a generator of <math>C_m\;</math> and
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| <math>a^rba^{-r}=b^{k^r}\;</math>, hence the presentation:
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| :<math>\langle a,\;b \mid a^m=e,\;b^n = e,\;aba^{-1}=b^{k^{r}}\;\rangle</math>
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| gives a group isomorphic to the previous one.
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| The [[fundamental group]] of the [[Klein bottle]] can be presented in the form
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| :<math>\langle a,\;b \mid aba^{-1}=b^{-1}\;\rangle</math>
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| and is therefore a semidirect product of the group of integers, <math>\mathbb{Z}</math>, with <math>\mathbb{Z}</math>. The corresponding homomorphism <math>\varphi : \mathbb{Z} \to \mathrm{Aut}(\mathbb{Z})</math> is given by <math>\varphi(h)(n)=(-1)^h n</math>.
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| The [[Euclidean group]] of all rigid motions ([[isometry|isometries]]) of the plane (maps ''f'' : '''R'''<sup>2</sup> → '''R'''<sup>2</sup> such that the Euclidean distance between ''x'' and ''y'' equals the distance between ''f''(''x'') and ''f''(''y'') for all ''x'' and ''y'' in '''R'''<sup>2</sup>) is isomorphic to a semidirect product of the abelian group '''R'''<sup>2</sup> (which describes translations) and the group O(2) of [[orthogonal matrix|orthogonal]] 2×2 matrices (which describes rotations and reflections that keep the origin fixed). Applying a translation and then a rotation or reflection has the same effect as applying the rotation or reflection first and then a translation by the rotated or reflected translation vector (i.e. applying the [[Conjugation of isometries in Euclidean space|conjugate]] of the original translation). This shows that the group of translations is a normal subgroup of the Euclidean group, that the Euclidean group is a semidirect product of the translation group and O(2), and that the corresponding homomorphism <math>\varphi : O(2) \to \mathrm{Aut}(\mathbb{R}^2)</math> is given by [[matrix multiplication]]: <math>\varphi(h)(n)=hn</math>.
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| The [[orthogonal group]] O(''n'') of all orthogonal real ''n''×''n'' matrices (intuitively the set of all rotations and reflections of ''n''-dimensional space which keep the origin fixed) is isomorphic to a semidirect product of the group SO(''n'') (consisting of all orthogonal matrices with [[determinant]] 1, intuitively the rotations of ''n''-dimensional space) and ''C''<sub>2</sub>. If we represent ''C''<sub>2</sub> as the multiplicative group of matrices {''I'', ''R''}, where ''R'' is a reflection of ''n'' dimensional space which keeps the origin fixed (i.e. an orthogonal matrix with determinant –1 representing an [[involution (mathematics)|involution]]), then φ : ''C''<sub>2</sub> → Aut(SO(''n'')) is given by φ(''H'')(''N'') = ''H'' ''N'' ''H''<sup>–1</sup> for all ''H'' in ''C''<sub>2</sub> and ''N'' in SO(''n''). In the non-trivial case ( ''H'' is not the identity) this means that φ(''H'') is conjugation of operations by the reflection (a rotation axis and the direction of rotation are replaced by their "mirror image").
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| The group of [[semilinear transformation]]s on a vector space ''V'' over a field <math>\mathbb{K}</math>, often denoted <math>\operatorname{\Gamma L}(V)</math>, is isomorphic to a semidirect product of the [[linear group]] <math>\operatorname{GL}(V)</math> (a [[normal subgroup]] of <math>\operatorname{\Gamma L}(V)</math>), and the [[automorphism group]] of <math>\mathbb{K}</math>.
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| == Relation to direct products ==
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| Suppose ''G'' is a semidirect product of the normal subgroup ''N'' and the subgroup ''H''. If ''H'' is also normal in ''G'', or equivalently, if there exists a homomorphism ''G'' → ''N'' which is the identity on ''N'', then ''G'' is the [[direct product of groups|direct product]] of ''N'' and ''H''.
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| The direct product of two groups ''N'' and ''H'' can be thought of as the semidirect product of ''N'' and ''H'' with respect to φ(''h'') = id<sub>''N''</sub> for all ''h'' in ''H''.
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| Note that in a direct product, the order of the factors is not important, since ''N'' × ''H'' is isomorphic to ''H'' × ''N''. This is not the case for semidirect products, as the two factors play different roles.
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| == Generalizations ==
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| The construction of semidirect products can be pushed much further. The [[Zappa–Szep product]] of groups is a generalization which, in its internal version, does not assume that either subgroup is normal. There is also a construction in [[ring theory]], the [[crossed product of rings]]. This is seen naturally as soon as one constructs a [[group ring]] for a semidirect product of groups. There is also the semidirect sum of [[Lie algebras]]. Given a [[group action]] on a [[topological space]], there is a corresponding crossed product which will in general be non-commutative even if the group is abelian. This kind of ring (see [[crossed product]] for a related construction) can play the role of the ''space of orbits'' of the group action, in cases where that space cannot be approached by conventional topological techniques – for example in the work of [[Alain Connes]] (cf. [[noncommutative geometry]]).
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| There are also far-reaching generalisations in [[category theory]]. They show how to construct ''[[fibred category|fibred categories]]'' from ''[[indexed category|indexed categories]]''. This is an abstract form of the outer semidirect product construction.
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| === Groupoids ===
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| Another generalisation is for groupoids. This occurs in topology because if a group <math>G</math> acts on a space <math>X</math> it also acts on the [[fundamental groupoid]] <math>\pi_1(X)</math> of the space. The semidirect product <math>\pi_1(X) \rtimes G</math> is then relevant to finding the fundamental groupoid of the [[orbit space]] <math>X/G</math>. For full details see Chapter 11 of the book referenced below, and also some details in semidirect product<ref>[http://ncatlab.org/nlab/show/semidirect+product Ncatlab.org]</ref> in [[nLab|ncatlab]].
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| === Abelian categories ===
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| Non-trivial semidirect products do ''not'' arise in [[abelian categories]], such as the [[category of modules]]. In this case, the [[splitting lemma]] shows that every semidirect product is a direct product. Thus the existence of semidirect products reflects a failure of the category to be abelian.
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| == Notation ==
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| Usually the semidirect product of a group ''H'' acting on a group ''N'' (in most cases by conjugation as subgroups of a common group) is denoted by <math>N\rtimes H</math> or <math>H\ltimes N</math>. However, some sources may use this symbol with the opposite meaning. In case the action <math>\phi :H \rightarrow \operatorname{Aut}(N)</math> should be made explicit, one also writes <math>N\rtimes_{\phi}H</math>. One way of thinking about the <math>N\rtimes H</math> symbol is as a combination of the symbol for normal subgroup (<math>\triangleleft</math>) and the symbol for the product (<math>\times</math>).
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| [[Unicode]] lists four variants:<ref>See [http://www.unicode.org/charts/symbols.htm unicode.org]</ref>
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| :{| cellspacing="0"
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| ! || || value || || MathML || || Unicode description
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| |-
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| | ⋉ || || U+22C9 || || ltimes || || LEFT NORMAL FACTOR SEMIDIRECT PRODUCT
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| | ⋊ || || U+22CA || || rtimes || || RIGHT NORMAL FACTOR SEMIDIRECT PRODUCT
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| |-
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| | ⋋ || || U+22CB || || lthree || || LEFT SEMIDIRECT PRODUCT
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| |-
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| | ⋌ || || U+22CC || || rthree || || RIGHT SEMIDIRECT PRODUCT
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| |}
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| Here the Unicode description of the rtimes symbol says "right normal factor", in contrast to its usual meaning in mathematical practice.
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| In [[LaTeX]], the commands \rtimes and \ltimes produce the corresponding characters.
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| == See also ==
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| *[[holomorph (mathematics)|Holomorph]]
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| *[[Subdirect product]]
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| *[[Wreath product]]
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| *[[Affine Lie algebra]]
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| ==Notes==
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| {{reflist}}
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| ==References==
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| {{refimprove|date=June 2009}}
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| * R. Brown, Topology and groupoids, Booksurge 2006. ISBN 1-4196-2722-8
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| [[Category:Group theory]]
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