Subgroup

{{short description|Subset of a group that forms a group itself}} {{other uses}} {{Group theory sidebar |Basics}}

In group theory, a branch of mathematics, a subset of a group G is a subgroup of G if the members of that subset form a group with respect to the group operation in G.

Formally, given a group {{mvar|G}} under a binary operation ∗, a subset {{mvar|H}} of {{mvar|G}} is called a '''subgroup''' of {{mvar|G}} if {{mvar|H}} also forms a group under the operation ∗. More precisely, {{mvar|H}} is a subgroup of {{mvar|G}} if the restriction of ∗ to {{math|''H'' × ''H''}} is a group operation on {{mvar|H}}. This is often denoted {{math|''H'' ≤ ''G''}}, read as "{{mvar|H}} is a subgroup of {{mvar|G}}".

The '''trivial subgroup''' of any group is the subgroup {''e''} consisting of just the identity element.{{sfn|Gallian|2013|p=61}}

A '''proper subgroup''' of a group {{mvar|G}} is a subgroup {{mvar|H}} which is a proper subset of {{mvar|G}} (that is, {{math|''H'' ≠ ''G''}}). This is often represented notationally by {{math|''H'' < ''G''}}, read as "{{mvar|H}} is a proper subgroup of {{mvar|G}}". Some authors also exclude the trivial group from being proper (that is, {{math|''H'' ≠ {''e''}{{0ws}}}}).{{sfn|Hungerford|1974|p=32}}{{sfn|Artin|2011|p=43}}

If {{mvar|H}} is a subgroup of {{mvar|G}}, then {{mvar|G}} is sometimes called an '''overgroup''' of {{mvar|H}}.

The same definitions apply more generally when {{mvar|G}} is an arbitrary semigroup, but this article will only deal with subgroups of groups.

==Subgroup tests==

Suppose that {{mvar|G}} is a group, and {{mvar|H}} is a subset of {{mvar|G}}. For now, assume that the group operation of {{mvar|G}} is written multiplicatively, denoted by juxtaposition. *Then {{mvar|H}} is a subgroup of {{mvar|G}} if and only if {{mvar|H}} is nonempty and closed under products and inverses. ''Closed under products'' means that for every {{mvar|a}} and {{mvar|b}} in {{mvar|H}}, the product {{mvar|ab}} is in {{mvar|H}}. ''Closed under inverses'' means that for every {{mvar|a}} in {{mvar|H}}, the inverse {{math|''a''^&minus;1}} is in {{mvar|H}}. These two conditions can be combined into one, that for every {{mvar|a}} and {{mvar|b}} in {{mvar|H}}, the element {{math|''ab''^&minus;1}} is in {{mvar|H}}, but it is more natural and usually just as easy to test the two closure conditions separately.{{sfn|Kurzweil|Stellmacher|1998|p=4}} *When {{mvar|H}} is ''finite'', the test can be simplified: {{mvar|H}} is a subgroup if and only if it is nonempty and closed under products. These conditions alone imply that every element {{mvar|a}} of {{mvar|H}} generates a finite cyclic subgroup of {{mvar|H}}, say of order {{mvar|n}}, and then the inverse of {{mvar|a}} is {{math|''a''^{''n''&minus;1}}}.{{sfn|Kurzweil|Stellmacher|1998|p=4}} If the group operation is instead denoted by addition, then ''closed under products'' should be replaced by ''closed under addition'', which is the condition that for every {{mvar|a}} and {{mvar|b}} in {{mvar|H}}, the sum {{math|''a'' + ''b''}} is in {{mvar|H}}, and ''closed under inverses'' should be edited to say that for every {{mvar|a}} in {{mvar|H}}, the inverse {{math|−''a''}} is in {{mvar|H}}.

==Basic properties of subgroups==

*The identity of a subgroup is the identity of the group: if {{mvar|G}} is a group with identity {{mvar|e_G}}, and {{mvar|H}} is a subgroup of {{mvar|G}} with identity {{mvar|e_H}}, then {{math|1=''e_H'' = ''e_G''}}. *The inverse of an element in a subgroup is the inverse of the element in the group: if {{mvar|H}} is a subgroup of a group {{mvar|G}}, and {{mvar|a}} and {{mvar|b}} are elements of {{mvar|H}} such that {{math|1=''ab'' = ''ba'' = ''e_H''}}, then {{math|1=''ab'' = ''ba'' = ''e_G''}}. *If {{mvar|H}} is a subgroup of {{mvar|G}}, then the inclusion map {{math|''H'' → ''G''}} sending each element {{mvar|a}} of {{mvar|H}} to itself is a homomorphism. *The intersection of subgroups {{mvar|A}} and {{mvar|B}} of {{mvar|G}} is again a subgroup of {{mvar|G}}.{{sfn|Jacobson|2009|p=41}} For example, the intersection of the {{mvar|x}}-axis and {{mvar|y}}-axis in {{tmath|\R^2}} under addition is the trivial subgroup. More generally, the intersection of an arbitrary collection of subgroups of {{mvar|G}} is a subgroup of {{mvar|G}}. *The union of subgroups {{mvar|A}} and {{mvar|B}} is a subgroup if and only if {{math|''A'' ⊆ ''B''}} or {{math|''B'' ⊆ ''A''}}. A non-example: {{tmath|2\Z \cup 3\Z}} is not a subgroup of {{tmath|\Z,}} because 2 and 3 are elements of this subset whose sum, 5, is not in the subset. Similarly, the union of the {{mvar|x}}-axis and the {{mvar|y}}-axis in {{tmath|\R^2}} is not a subgroup of {{tmath|\R^2.}} *If {{mvar|S}} is a subset of {{mvar|G}}, then there exists a smallest subgroup containing {{mvar|S}}, namely the intersection of all of subgroups containing {{mvar|S}}; it is denoted by {{math|{{angbr|''S''}}}} and is called the subgroup generated by {{mvar|S}}. An element of {{mvar|G}} is in {{math|{{angbr|''S''}}}} if and only if it is a finite product of elements of {{mvar|S}} and their inverses, possibly repeated.{{sfn|Ash|2002}} *Every element {{mvar|a}} of a group {{mvar|G}} generates a cyclic subgroup {{math|{{angbr|''a''}}}}. If {{math|{{angbr|''a''}}}} is isomorphic to {{tmath|\Z/n\Z}} (the integers {{math|mod ''n''}}) for some positive integer {{mvar|n}}, then {{mvar|n}} is the smallest positive integer for which {{math|1=''aⁿ'' = ''e''}}, and {{mvar|n}} is called the ''order'' of {{mvar|a}}. If {{math|{{angbr|''a''}}}} is isomorphic to {{tmath|\Z,}} then {{mvar|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 {{mvar|e}} is the identity of {{mvar|G}}, then the trivial group {{math|{''e''} }} is the minimum subgroup of {{mvar|G}}, while the maximum subgroup is the group {{mvar|G}} itself.

[[File:Left cosets of Z 2 in Z 8.svg|thumb|{{mvar|G}} is the group <math>\Z/8\Z,</math> the integers mod 8 under addition. The subgroup {{mvar|H}} contains only 0 and 4, and is isomorphic to <math>\Z/2\Z.</math> There are four left cosets of {{mvar|H}}: {{mvar|H}} itself, {{math|1 + ''H''}}, {{math|2 + ''H''}}, and {{math|3 + ''H''}} (written using additive notation since this is an additive group). Together they partition the entire group {{mvar|G}} into equal-size, non-overlapping sets. The index {{math|[''G'' : ''H'']}} is 4.]]

==Cosets and Lagrange's theorem== {{Main|Coset|Lagrange's theorem (group theory)}} Given a subgroup {{mvar|H}} and some {{mvar|a}} in {{mvar|G}}, we define the '''left coset''' {{math|1=''aH'' = {''ah'' : ''h'' in ''H''}.}} Because {{mvar|a}} is invertible, the map {{math|φ : ''H'' → ''aH''}} given by {{math|1=φ(''h'') = ''ah''}} is a bijection. Furthermore, every element of {{mvar|G}} is contained in precisely one left coset of {{mvar|H}}; the left cosets are the equivalence classes corresponding to the equivalence relation {{math|''a''₁ ~ ''a''₂}} if and only if {{tmath|a_1^{-1}a_2}} is in {{mvar|H}}. The number of left cosets of {{mvar|H}} is called the index of {{mvar|H}} in {{mvar|G}} and is denoted by {{math|[''G'' : ''H'']}}.

Lagrange's theorem states that for a finite group {{mvar|G}} and a subgroup {{mvar|H}}, : <math> [ G : H ] = { |G| \over |H| }</math> where {{mvar|{{abs|G}}}} and {{mvar|{{abs|H}}}} denote the orders of {{mvar|G}} and {{mvar|H}}, respectively. In particular, the order of every subgroup of {{mvar|G}} (and the order of every element of {{mvar|G}}) must be a divisor of {{mvar|{{abs|G}}}}.<ref>See a [https://www.youtube.com/watch?v=TCcSZEL_3CQ didactic proof in this video].</ref>{{sfn|Dummit|Foote|2004|p=90}}

'''Right cosets''' are defined analogously: {{math|1=''Ha'' = {''ha'' : ''h'' in ''H''}.}} They are also the equivalence classes for a suitable equivalence relation and their number is equal to {{math|[''G'' : ''H'']}}.

If {{math|1=''aH'' = ''Ha''}} for every {{mvar|a}} in {{mvar|G}}, then {{mvar|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. More generally, if {{mvar|p}} is the lowest prime dividing the order of a finite group {{mvar|G}}, then any subgroup of index {{mvar|p}} (if such exists) is normal.

==Example: Subgroups of Z₈==<!-- This section is linked from List of small groups --> Let {{mvar|G}} be the finite cyclic group :<math>\mathrm{Z}_8 = \{0,1,2,3,4,5,6,7\}</math> under addition modulo 8. The subset <math>\{0,2,4,6\}</math> consisting of multiples of 2 is a subgroup of <math>\mathrm{Z}_8</math>. More generally, for each divisor {{mvar|d}} of 8, the multiples of {{mvar|d}} form a subgroup. Explicitly, for <math>d=1,2,4,8</math>, these subgroups are <math>\{0,1,2,3,4,5,6,7\}, \{0,2,4,6\}, \{0,4\}, \{0\}</math>.

In general, for any positive integer {{mvar|n}}, one can describe all subgroups of the finite cyclic group <math>\mathrm{Z}_n</math> similarly: for each divisor {{mvar|d}} of {{mvar|n}}, the multiples of {{mvar|d}} in <math>\mathrm{Z}_n</math> form a subgroup of order <math>n/d</math>, and every subgroup arises in this way.

Subgroups of cyclic groups are cyclic.{{sfn|Gallian|2013|p=81}}

==Example: Subgroups of S₄{{anchor|Subgroups of S4}}==

The symmetric group {{math|S₄}} is the group whose elements are the permutations of <math>\{1,2,3,4\}</math>.<br> Below are all its subgroups, ordered by cardinality.<br>

{| style="width:100%" | style="vertical-align:top;"| {{multiple image | align = right | image1 = Symmetric group S4; lattice of subgroups Hasse diagram; all 30 subgroups.svg | width1 = 250 | caption1 = All 30 subgroups | image2 = Symmetric group S4; lattice of subgroups Hasse diagram; 11 different cycle graphs.svg | width2 = 185 | caption2 = Simplified | footer = Hasse diagrams of the lattice of subgroups of {{math|S₄}} }} |}

===24 elements=== Like each group, {{math|S₄}} is a subgroup of itself.

===12 elements=== The alternating group {{math|A₄}} consists of all the even permutations in {{math|S₄}}. Since it is of index 2, it is a normal subgroup.

===8 elements=== There are three subgroups of order 8, each isomorphic to the dihedral group {{math|D₄}}, the group of symmetries of a square.

Labeling the vertices of a square <math>1,2,3,4</math> clockwise lets one view {{math|D₄}} as a subgroup of {{math|S₄}}. This subgroup is generated by the 90-degree clockwise rotation and by the reflection in the diagonal axis joining vertices 1 and 3; these are the permutations <math>(1234)</math> and <math>(24)</math>.

Up to symmetries of the square, there are three different ways to label the vertices of a square, distinguished by which pairs of numbers appear on opposite corners. In the labeling above, 1 and 3 were opposite, and 2 and 4 were opposite; another choice has 1 and 4 opposite, and 2 and 3 opposite; the third choice has 1 and 2 opposite, and 3 and 4 opposite. The three labelings give rise to three different subgroups of order 8 in {{math|S₄}}, conjugate to each other, each isomorphic to {{math|D₄}}.

===6 elements=== There are four subgroups of order 6, each isomorphic to {{math|S₃}}. Each is the stabilizer of one of the elements of <math>\{1,2,3,4\}</math>. For example, the stabilizer of 4 is the group of permutations in {{math|S₄}} that map 4 to 4, while permuting <math>\{1,2,3\}</math> in an arbitrary way; it is generated by the permutations <math>(12)</math> and <math>(123)</math>, for instance. The four subgroups of order 6 are conjugate to each other.

===4 elements=== There are seven subgroups of order 4, falling into three conjugacy classes of subgroups:

*The subset <math>\{1,(12)(34),(13)(24),(14)(23)\}</math> is a normal subgroup isomorphic to the Klein four-group {{math|V₄}}.

*The group generated by <math>(12)</math> and <math>(34)</math> is another subgroup isomorphic to {{math|V₄}}, but it is not normal. Instead it has conjugates, namely the group generated by <math>(13)</math> and <math>(24)</math> and the group generated by <math>(14)</math> and <math>(23)</math>.

*Each of the six 4-cycles in {{math|S₄}} generates a cyclic subgroup of order 4, but each 4-cycle generates the same subgroup as its inverse, so there are only three distinct subgroups of this type. These three subgroups are conjugate to each other because all 4-cycles in {{math|S₄}} are conjugate to each other.

===3 elements=== There are four subgroups of order 3, each generated by a 3-cycle. There are eight 3-cycles in {{math|S₄}}, but each generates the same subgroup as its inverse. The resulting four subgroups are conjugate to each other.

===2 elements=== There are nine subgroups of order 2, falling into two conjugacy classes of subgroups:

*Each of the <math>\binom{4}{2} = 6</math> transpositions (2-cycles) generates a subgroup of order 2. These six subgroups are conjugate.

*Each of the double-transpositions <math>(12)(34)</math>, <math>(13)(24)</math>, <math>(14)(23)</math> generates a subgroup of order 2. These three subgroups are conjugate.

===1 element=== The trivial subgroup is the unique subgroup of order 1.

==Other examples== *The even integers form a subgroup {{tmath|2\Z}} of the integer ring {{tmath|\Z:}} the sum of two even integers is even, and the negative of an even integer is even. *Every ideal in a ring {{mvar|R}} is a subgroup of the additive group of {{mvar|R}}. *Every linear subspace of a vector space is a subgroup of the additive group of vectors. *In an abelian group, the elements of finite order form a subgroup called the torsion subgroup.

== Notes == <references/>

== References == * {{Citation| last=Jacobson| first=Nathan | author-link=Nathan Jacobson| year=2009| title=Basic algebra| edition=2nd| volume = 1 | publisher=Dover| isbn = 978-0-486-47189-1}}. * {{Citation| last=Hungerford| first=Thomas| author-link=Thomas W. Hungerford| year=1974| title=Algebra| edition=1st| publisher=Springer-Verlag| isbn =9780387905181}}. * {{Citation| last=Artin| first=Michael| author-link=Michael Artin| year=2011| title=Algebra| edition=2nd| publisher=Prentice Hall| isbn = 9780132413770}}. * {{Cite book|title=Abstract algebra|last1=Dummit|first1=David S.|last2=Foote|first2=Richard M.|date=2004|publisher=Wiley|isbn=9780471452348|edition=3rd|location=Hoboken, NJ|oclc=248917264}} * {{Cite book |last=Gallian |first=Joseph A. | author-link=Joseph Gallian|title=Contemporary abstract algebra |date=2013 |publisher=Brooks/Cole Cengage Learning |isbn=978-1-133-59970-8 |edition=8th |location=Boston, MA |oclc=807255720}} * {{Cite book|last1=Kurzweil|first1=Hans|last2=Stellmacher|first2=Bernd|date=1998|title=Theorie der endlichen Gruppen|url=http://dx.doi.org/10.1007/978-3-642-58816-7|series=Springer-Lehrbuch|doi=10.1007/978-3-642-58816-7|isbn=978-3-540-60331-3 }} * {{Cite book |last=Ash |first=Robert B. |url=https://faculty.math.illinois.edu/~r-ash/Algebra.html |title=Abstract Algebra: The Basic Graduate Year |date=2002 |publisher=Department of Mathematics University of Illinois |language=en}}

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