User talk:67.86.48.64 and Subgroup: Difference between pages
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{{Basic notions in group theory}} |
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In [[group theory]], given a [[group (mathematics)|group]] ''G'' under a [[binary operation]] *, we say that some [[subset]] ''H'' of ''G'' is a '''subgroup''' of ''G'' if ''H'' also forms a group under the operation *. More precisely, ''H'' is a subgroup of ''G'' if the [[function (mathematics)#Restrictions and extensions|restriction]] of * to ''H x H'' is a group operation on ''H''. This is usually represented notationally by ''H'' ≤ ''G'', read as "''H'' is a subgroup of ''G''". |
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A '''proper subgroup''' of a group ''G'' is a subgroup ''H'' which is a [[subset|proper subset]] of ''G'' (i.e. ''H'' ≠ ''G''). The '''trivial subgroup''' of any group is the subgroup {''e''} consisting of just the identity element. If ''H'' is a subgroup of ''G'', then ''G'' is sometimes called an ''overgroup'' of ''H''. |
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The same definitions apply more generally when ''G'' is an arbitrary [[semigroup]], but this article will only deal with subgroups of groups. The group ''G'' is sometimes denoted by the ordered pair (''G'',*), usually to emphasize the operation * when ''G'' carries multiple algebraic or other structures. |
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In the following, we follow the usual convention of dropping * and writing the product ''a''*''b'' as simply ''ab''. |
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==Basic properties of subgroups== |
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*''H'' is a subgroup of the group ''G'' if and only if it is nonempty and closed under products and inverses. (The closure conditions mean the following: whenever ''a'' and ''b'' are in ''H'', then ''ab'' and ''a''<sup>−1</sup> are also in ''H''. These two conditions can be combined into one equivalent condition: whenever ''a'' and ''b'' are in ''H'', then ''ab''<sup>−1</sup> is also in ''H''.) In the case that ''H'' is finite, then ''H'' is a subgroup [[if and only if]] ''H'' is closed under products. (In this case, every element ''a'' of ''H'' generates a finite cyclic subgroup of ''H'', and the inverse of ''a'' is then ''a''<sup>−1</sup> = ''a''<sup>''n'' − 1</sup>, where ''n'' is the order of ''a''.) |
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*The above condition can be stated in terms of a [[homomorphism]]; that is, ''H'' is a subgroup of a group ''G'' if and only if ''H'' is a subset of ''G'' and there is an inclusion homomorphism (i.e., i(''a'') = ''a'' for every ''a'') from ''H'' to ''G''. |
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*The identity of a subgroup is the identity of the group: if ''G'' is a group with identity ''e''<sub>''G''</sub>, and ''H'' is a subgroup of ''G'' with identity ''e''<sub>''H''</sub>, then ''e''<sub>''H''</sub> = ''e''<sub>''G''</sub>. |
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*The inverse of an element in a subgroup is the inverse of the element in the group: if ''H'' is a subgroup of a group ''G'', and ''a'' and ''b'' are elements of ''H'' such that ''ab'' = ''ba'' = ''e''<sub>''H''</sub>, then ''ab'' = ''ba'' = ''e''<sub>''G''</sub>. |
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*The intersection of subgroups ''A'' and ''B'' is again a subgroup. The union of subgroups ''A'' and ''B'' is a subgroup if and only if either ''A'' or ''B'' contains the other, since for example 2 and 3 are in the union of 2Z and 3Z but their sum 5 is not. Another example is the union of the x-axis and the y-axis in the plane (with the addition operation); each of these objects is a subgroup but their union is not. This also serves as an example of two subgroups, whose intersection is precisely the identity. |
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*If ''S'' is a subset of ''G'', then there exists a minimum subgroup containing ''S'', which can be found by taking the intersection of all of subgroups containing ''S''; it is denoted by <''S''> and is said to be the [[generating set of a group|subgroup generated by ''S'']]. An element of ''G'' is in <''S''> if and only if it is a finite product of elements of ''S'' and their inverses. |
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*Every element ''a'' of a group ''G'' generates the cyclic subgroup <''a''>. If <''a''> is [[group isomorphism|isomorphic]] to '''Z'''/''n'''''Z''' for some positive integer ''n'', then ''n'' is the smallest positive integer for which ''a''<sup>''n''</sup> = ''e'', and ''n'' is called the ''order'' of ''a''. If <''a''> is isomorphic to '''Z''', then ''a'' is said to have ''infinite order''. |
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*The subgroups of any given group form a [[complete lattice]] under inclusion, called the [[lattice of subgroups]]. (While the [[infimum]] here is the usual set-theoretic intersection, the [[supremum]] of a set of subgroups is the subgroup ''generated by'' the set-theoretic union of the subgroups, not the set-theoretic union itself.) If ''e'' is the identity of ''G'', then the trivial group {''e''} is the [[partial order|minimum]] subgroup of ''G'', while the [[partial order|maximum]] subgroup is the group ''G'' itself. |
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==Example==<!-- This section is linked from [[List of small groups]] --> |
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Let ''G'' be the [[abelian group]] whose elements are |
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:''G''={0,2,4,6,1,3,5,7} |
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and whose group operation is [[modular arithmetic|addition modulo eight]]. Its [[Cayley table]] is |
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{| border="2" cellpadding="7" |
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!style="background:#FFFFAA;"| + |
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!style="background:#FFFFAA;"| <font color="red">0 |
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!style="background:#FFFFAA;"| <font color="red">2 |
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!style="background:#FFFFAA;"| <font color="red">4 |
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!style="background:#FFFFAA;"| <font color="red">6 |
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!style="background:#FFFFAA;"| <font color="blue">1 |
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!style="background:#FFFFAA;"| <font color="blue">3 |
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!style="background:#FFFFAA;"| <font color="blue">5 |
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!style="background:#FFFFAA;"| <font color="blue">7 |
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|- |
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!style="background:#FFFFAA;"| <font color="red">0 |
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| <font color="orange">0 || <font color="red">2 || <font color="orange">4 || <font color="red">6 || <font color="blue">1 || <font color="blue">3 || <font color="blue">5 || <font color="blue">7 |
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|- |
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!style="background:#FFFFAA;"| <font color="red">2 |
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| <font color="red">2 || <font color="red">4 || <font color="red">6 || <font color="red">0 || <font color="blue">3 || <font color="blue">5 || <font color="blue">7 || <font color="blue">1 |
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|- |
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!style="background:#FFFFAA;"| <font color="red">4 |
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| <font color="orange">4 || <font color="red">6 || <font color="orange">0 || <font color="red">2 || <font color="blue">5 || <font color="blue">7 || <font color="blue">1 || <font color="blue">3 |
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|- |
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!style="background:#FFFFAA;"| <font color="red">6 |
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| <font color="red">6 || <font color="red">0 || <font color="red">2 || <font color="red">4 || <font color="blue">7 || <font color="blue">1 || <font color="blue">3 || <font color="blue">5 |
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|- |
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!style="background:#FFFFAA;"| <font color="blue">1 |
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| <font color="blue">1 || <font color="blue">3 || <font color="blue">5 || <font color="blue">7 || <font color="red">2 || <font color="red">4 || <font color="red">6 || <font color="red">0 |
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|- |
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!style="background:#FFFFAA;"| <font color="blue">3 |
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| <font color="blue">3 || <font color="blue">5 || <font color="blue">7 || <font color="blue">1 || <font color="red">4 || <font color="red">6 || <font color="red">0 || <font color="red">2 |
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|- |
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!style="background:#FFFFAA;"| <font color="blue">5 |
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| <font color="blue">5 || <font color="blue">7 || <font color="blue">1 || <font color="blue">3 || <font color="red">6 || <font color="red">0 || <font color="red">2 || <font color="red">4 |
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|- |
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!style="background:#FFFFAA;"| <font color="blue">7 |
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| <font color="blue">7 || <font color="blue">1 || <font color="blue">3 || <font color="blue">5 || <font color="red">0 || <font color="red">2 || <font color="red">4 || <font color="red">6 |
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|} |
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This group has a pair of nontrivial subgroups: <font color="orange">''J''={0,4}</font color> and <font color="red">''H''={0,2,4,6}</font color>, where ''J'' is also a subgroup of ''H''. The Cayley table for ''H'' is the top-left quadrant of the Cayley table for ''G''. The group ''G'' is [[cyclic group|cyclic]], and so are its subgroups. In general, subgroups of cyclic groups are also cyclic. |
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==Cosets and Lagrange's theorem== |
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Given a subgroup ''H'' and some ''a'' in G, we define the '''left [[coset]]''' ''aH'' = {''ah'' : ''h'' in ''H''}. Because ''a'' is invertible, the map φ : ''H'' → ''aH'' given by φ(''h'') = ''ah'' is a [[bijection]]. Furthermore, every element of ''G'' is contained in precisely one left coset of ''H''; the left cosets are the equivalence classes corresponding to the [[equivalence relation]] ''a''<sub>1</sub> ~ ''a''<sub>2</sub> [[if and only if]] ''a''<sub>1</sub><sup>−1</sup>''a''<sub>2</sub> is in ''H''. The number of left cosets of ''H'' is called the ''index'' of ''H'' in ''G'' and is denoted by [''G'' : ''H'']. |
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[[Lagrange's theorem (group theory)|Lagrange's theorem]] states that for a finite group ''G'' and a subgroup ''H'', |
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:<math> [ G : H ] = { |G| \over |H| } </math> |
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where ''|G|'' and ''|H|'' denote the [[order (group theory)|order]]s of ''G'' and ''H'', respectively. In particular, the order of every subgroup of ''G'' (and the order of every element of ''G'') must be a [[divisor]] of ''|G|''. |
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'''Right cosets''' are defined analogously: ''Ha'' = {''ha'' : ''h'' in ''H''}. They are also the equivalence classes for a suitable equivalence relation and their number is equal to [''G'' : ''H'']. |
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If ''aH'' = ''Ha'' for every ''a'' in ''G'', then ''H'' is said to be a [[normal subgroup]]. Every subgroup of index 2 is normal: the left cosets, and also the right cosets, are simply the subgroup and its complement. |
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== See also == |
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* [[Cartan subgroup]] |
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* [[Fitting subgroup]] |
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* [[stable subgroup]] |
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[[Category:Group theory]] |
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[[Category:Subgroup properties]] |
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[[cs:Podgrupa]] |
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[[da:Undergruppe]] |
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[[de:Untergruppe]] |
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[[es:Subgrupo]] |
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[[fr:Sous-groupe]] |
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[[ko:부분군]] |
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[[hr:Podgrupa]] |
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[[it:Sottogruppo]] |
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[[nl:Ondergroep (wiskunde)]] |
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[[pl:Podgrupa]] |
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[[pt:Subgrupo]] |
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[[ru:Подгруппа]] |
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[[sr:Подгрупа (математика)]] |
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[[fi:Aliryhmä]] |
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[[vi:Nhóm con]] |
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[[tr:Altöbek]] |
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[[zh:子群]] |
Revision as of 04:06, 10 October 2008
Algebraic structure → Group theory Group theory |
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In group theory, given a group G under a binary operation *, we say that some subset H of G is a subgroup of G if H also forms a group under the operation *. More precisely, H is a subgroup of G if the restriction of * to H x H is a group operation on H. This is usually represented notationally by H ≤ G, read as "H is a subgroup of G".
A proper subgroup of a group G is a subgroup H which is a proper subset of G (i.e. H ≠ G). The trivial subgroup of any group is the subgroup {e} consisting of just the identity element. If H is a subgroup of G, then G is sometimes called an overgroup of H.
The same definitions apply more generally when G is an arbitrary semigroup, but this article will only deal with subgroups of groups. The group G is sometimes denoted by the ordered pair (G,*), usually to emphasize the operation * when G carries multiple algebraic or other structures.
In the following, we follow the usual convention of dropping * and writing the product a*b as simply ab.
Basic properties of subgroups
- H is a subgroup of the group G if and only if it is nonempty and closed under products and inverses. (The closure conditions mean the following: whenever a and b are in H, then ab and a−1 are also in H. These two conditions can be combined into one equivalent condition: whenever a and b are in H, then ab−1 is also in H.) In the case that H is finite, then H is a subgroup if and only if H is closed under products. (In this case, every element a of H generates a finite cyclic subgroup of H, and the inverse of a is then a−1 = an − 1, where n is the order of a.)
- The above condition can be stated in terms of a homomorphism; that is, H is a subgroup of a group G if and only if H is a subset of G and there is an inclusion homomorphism (i.e., i(a) = a for every a) from H to G.
- The identity of a subgroup is the identity of the group: if G is a group with identity eG, and H is a subgroup of G with identity eH, then eH = eG.
- The inverse of an element in a subgroup is the inverse of the element in the group: if H is a subgroup of a group G, and a and b are elements of H such that ab = ba = eH, then ab = ba = eG.
- The intersection of subgroups A and B is again a subgroup. The union of subgroups A and B is a subgroup if and only if either A or B contains the other, since for example 2 and 3 are in the union of 2Z and 3Z but their sum 5 is not. Another example is the union of the x-axis and the y-axis in the plane (with the addition operation); each of these objects is a subgroup but their union is not. This also serves as an example of two subgroups, whose intersection is precisely the identity.
- If S is a subset of G, then there exists a minimum subgroup containing S, which can be found by taking the intersection of all of subgroups containing S; it is denoted by <S> and is said to be the subgroup generated by S. An element of G is in <S> if and only if it is a finite product of elements of S and their inverses.
- Every element a of a group G generates the cyclic subgroup <a>. If <a> is isomorphic to Z/nZ for some positive integer n, then n is the smallest positive integer for which an = e, and n is called the order of a. If <a> is isomorphic to Z, then a is said to have infinite order.
- The subgroups of any given group form a complete lattice under inclusion, called the lattice of subgroups. (While the infimum here is the usual set-theoretic intersection, the supremum of a set of subgroups is the subgroup generated by the set-theoretic union of the subgroups, not the set-theoretic union itself.) If e is the identity of G, then the trivial group {e} is the minimum subgroup of G, while the maximum subgroup is the group G itself.
Example
Let G be the abelian group whose elements are
- G={0,2,4,6,1,3,5,7}
and whose group operation is addition modulo eight. Its Cayley table is
+ | 0 | 2 | 4 | 6 | 1 | 3 | 5 | 7 |
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0 | 0 | 2 | 4 | 6 | 1 | 3 | 5 | 7 |
2 | 2 | 4 | 6 | 0 | 3 | 5 | 7 | 1 |
4 | 4 | 6 | 0 | 2 | 5 | 7 | 1 | 3 |
6 | 6 | 0 | 2 | 4 | 7 | 1 | 3 | 5 |
1 | 1 | 3 | 5 | 7 | 2 | 4 | 6 | 0 |
3 | 3 | 5 | 7 | 1 | 4 | 6 | 0 | 2 |
5 | 5 | 7 | 1 | 3 | 6 | 0 | 2 | 4 |
7 | 7 | 1 | 3 | 5 | 0 | 2 | 4 | 6 |
This group has a pair of nontrivial subgroups: J={0,4} and H={0,2,4,6}, where J is also a subgroup of H. The Cayley table for H is the top-left quadrant of the Cayley table for G. The group G is cyclic, and so are its subgroups. In general, subgroups of cyclic groups are also cyclic.
Cosets and Lagrange's theorem
Given a subgroup H and some a in G, we define the left coset aH = {ah : h in H}. Because a is invertible, the map φ : H → aH given by φ(h) = ah is a bijection. Furthermore, every element of G is contained in precisely one left coset of H; the left cosets are the equivalence classes corresponding to the equivalence relation a1 ~ a2 if and only if a1−1a2 is in H. The number of left cosets of H is called the index of H in G and is denoted by [G : H].
Lagrange's theorem states that for a finite group G and a subgroup H,
where |G| and |H| denote the orders of G and H, respectively. In particular, the order of every subgroup of G (and the order of every element of G) must be a divisor of |G|.
Right cosets are defined analogously: Ha = {ha : h in H}. They are also the equivalence classes for a suitable equivalence relation and their number is equal to [G : H].
If aH = Ha for every a in G, then H is said to be a normal subgroup. Every subgroup of index 2 is normal: the left cosets, and also the right cosets, are simply the subgroup and its complement.