In group theory, a branch of mathematics, given a groupG under a binary operation ∗, a subsetH of G is called 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 × H is a group operation on H. This is often denoted H ≤ G, read as "H is a subgroup of G".
The trivial subgroup of any group is the subgroup {e} consisting of just the identity element.[1]
A proper subgroup of a group G is a subgroup H which is a proper subset of G (that is, H ≠ G). This is often represented notationally by H < G, read as "H is a proper subgroup of G". Some authors also exclude the trivial group from being proper (that is, H ≠ {e}).[2][3]
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.
Suppose that G is a group, and H is a subset of G. For now, assume that the group operation of G is written multiplicatively, denoted by juxtaposition.
Then H is a subgroup of Gif and only ifH is nonempty and closed under products and inverses. Closed under products means that for every a and b in H, the product ab is in H. Closed under inverses means that for every a in H, the inverse a−1 is in H. These two conditions can be combined into one, that for every a and b in H, the element ab−1 is in H, but it is more natural and usually just as easy to test the two closure conditions separately.[4]
When H is finite, the test can be simplified: H is a subgroup if and only if it is nonempty and closed under products. These conditions alone imply that every element a of H generates a finite cyclic subgroup of H, say of order n, and then the inverse of a is an−1.[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 a and b in H, the sum a+b is in H, and closed under inverses should be edited to say that for every a in H, the inverse −a is in H.
Basic properties of subgroupsEdit
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.
If H is a subgroup of G, then the inclusion map H → G sending each element a of H to itself is a homomorphism.
The intersection of subgroups A and B of G is again a subgroup of G.[5] For example, the intersection of the x-axis and y-axis in R2 under addition is the trivial subgroup. More generally, the intersection of an arbitrary collection of subgroups of G is a subgroup of G.
The union of subgroups A and B is a subgroup if and only if A ⊆ B or B ⊆ A. A non-example: 2Z ∪ 3Z is not a subgroup of Z, because 2 and 3 are elements of this subset whose sum, 5, is not in the subset. Similarly, the union of the x-axis and the y-axis in R2 is not a subgroup of R2.
If S is a subset of G, then there exists a smallest subgroup containing S, namely the intersection of all of subgroups containing S; it is denoted by ⟨S⟩ and is called 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, possibly repeated.[6]
Every element a of a group G generates a cyclic subgroup ⟨a⟩. If ⟨a⟩ is isomorphic to Z/nZ (the integers mod n) 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.
G is the group , the integers mod 8 under addition. The subgroup H contains only 0 and 4, and is isomorphic to . There are four left cosets of H: H itself, 1+H, 2+H, and 3+H (written using additive notation since this is an additive group). Together they partition the entire group G into equal-size, non-overlapping sets. The index [G : H] is 4.
Given a subgroup H and some a in G, we define the left cosetaH = {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 relationa1 ~ a2if and only ifa1−1a2 is in H. The number of left cosets of H is called the index of H in G and is denoted by [G : H].
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|.[7][8]
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. More generally, if p is the lowest prime dividing the order of a finite group G, then any subgroup of index p (if such exists) is normal.
This group has two nontrivial subgroups: ■J = {0, 4} and ■H = {0, 4, 2, 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 Cayley table for J is the top-left quadrant of the Cayley table for H. The group G is cyclic, and so are its subgroups. In general, subgroups of cyclic groups are also cyclic.[9]
Example: Subgroups of S4Edit
S4 is the symmetric group whose elements correspond to the permutations of 4 elements. Below are all its subgroups, ordered by cardinality. Each group (except those of cardinality 1 and 2) is represented by its Cayley table.
Ash, Robert B. (2002). Abstract Algebra: The Basic Graduate Year. Department of Mathematics University of Illinois.
October 21, 2023
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For other uses see Subgroup disambiguation In group theory a branch of mathematics given a group G under a binary operation a subset H of G is called 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 H is a group operation on H This is often denoted H G read as H is a subgroup of G The trivial subgroup of any group is the subgroup e consisting of just the identity element 1 A proper subgroup of a group G is a subgroup H which is a proper subset of G that is H G This is often represented notationally by H lt G read as H is a proper subgroup of G Some authors also exclude the trivial group from being proper that is H e 2 3 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 Contents 1 Subgroup tests 2 Basic properties of subgroups 3 Cosets and Lagrange s theorem 4 Example Subgroups of Z8 5 Example Subgroups of S4 5 1 24 elements 5 2 12 elements 5 3 8 elements 5 4 6 elements 5 5 4 elements 5 6 3 elements 5 7 2 elements 5 8 1 element 6 Other examples 7 See also 8 Notes 9 ReferencesSubgroup tests EditSuppose that G is a group and H is a subset of G For now assume that the group operation of G is written multiplicatively denoted by juxtaposition Then H is a subgroup of G if and only if H is nonempty and closed under products and inverses Closed under products means that for every a and b in H the product ab is in H Closed under inverses means that for every a in H the inverse a 1 is in H These two conditions can be combined into one that for every a and b in H the element ab 1 is in H but it is more natural and usually just as easy to test the two closure conditions separately 4 When H is finite the test can be simplified H is a subgroup if and only if it is nonempty and closed under products These conditions alone imply that every element a of H generates a finite cyclic subgroup of H say of order n and then the inverse of a is an 1 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 a and b in H the sum a b is in H and closed under inverses should be edited to say that for every a in H the inverse a is in H Basic properties of subgroups EditThe 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 If H is a subgroup of G then the inclusion map H G sending each element a of H to itself is a homomorphism The intersection of subgroups A and B of G is again a subgroup of G 5 For example the intersection of the x axis and y axis in R2 under addition is the trivial subgroup More generally the intersection of an arbitrary collection of subgroups of G is a subgroup of G The union of subgroups A and B is a subgroup if and only if A B or B A A non example 2Z 3Z is not a subgroup of Z because 2 and 3 are elements of this subset whose sum 5 is not in the subset Similarly the union of the x axis and the y axis in R2 is not a subgroup of R2 If S is a subset of G then there exists a smallest subgroup containing S namely the intersection of all of subgroups containing S it is denoted by S and is called 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 possibly repeated 6 Every element a of a group G generates a cyclic subgroup a If a is isomorphic to Z nZ the integers mod n 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 nbsp G is the group Z 8 Z displaystyle mathbb Z 8 mathbb Z nbsp the integers mod 8 under addition The subgroup H contains only 0 and 4 and is isomorphic to Z 2 Z displaystyle mathbb Z 2 mathbb Z nbsp There are four left cosets of H H itself 1 H 2 H and 3 H written using additive notation since this is an additive group Together they partition the entire group G into equal size non overlapping sets The index G H is 4 Cosets and Lagrange s theorem EditMain articles Coset and Lagrange s theorem group theory 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 f H aH given by f 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 G H G H displaystyle G H G over H nbsp 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 7 8 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 More generally if p is the lowest prime dividing the order of a finite group G then any subgroup of index p if such exists is normal Example Subgroups of Z8 EditLet G be the cyclic group Z8 whose elements are G 0 4 2 6 1 5 3 7 displaystyle G left 0 4 2 6 1 5 3 7 right nbsp and whose group operation is addition modulo 8 Its Cayley table is 0 4 2 6 1 5 3 70 0 4 2 6 1 5 3 74 4 0 6 2 5 1 7 32 2 6 4 0 3 7 5 16 6 2 0 4 7 3 1 51 1 5 3 7 2 6 4 05 5 1 7 3 6 2 0 43 3 7 5 1 4 0 6 27 7 3 1 5 0 4 2 6This group has two nontrivial subgroups J 0 4 and H 0 4 2 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 Cayley table for J is the top left quadrant of the Cayley table for H The group G is cyclic and so are its subgroups In general subgroups of cyclic groups are also cyclic 9 Example Subgroups of S4 EditS4 is the symmetric group whose elements correspond to the permutations of 4 elements Below are all its subgroups ordered by cardinality Each group except those of cardinality 1 and 2 is represented by its Cayley table 24 elements Edit Like each group S4 is a subgroup of itself nbsp Symmetric group S4 nbsp All 30 subgroups nbsp SimplifiedHasse diagrams of the lattice of subgroups of S412 elements Edit The alternating group contains only the even permutations It is one of the two nontrivial proper normal subgroups of S4 The other one is its Klein subgroup nbsp Alternating group A4Subgroups nbsp nbsp nbsp nbsp nbsp 8 elements Edit nbsp Dihedral group of order 8Subgroups nbsp nbsp nbsp nbsp Dihedral group of order 8Subgroups nbsp nbsp nbsp nbsp Dihedral group of order 8Subgroups nbsp nbsp nbsp 6 elements Edit nbsp Symmetric group S3Subgroup nbsp nbsp Symmetric group S3Subgroup nbsp nbsp Symmetric group S3Subgroup nbsp nbsp Symmetric group S3Subgroup nbsp 4 elements Edit nbsp Klein four group nbsp Klein four group nbsp Klein four group nbsp Klein four group normal subgroup nbsp Cyclic group Z4 nbsp Cyclic group Z4 nbsp Cyclic group Z4 3 elements Edit nbsp Cyclic group Z3 nbsp Cyclic group Z3 nbsp Cyclic group Z3 nbsp Cyclic group Z3 2 elements Edit Each permutation p of order 2 generates a subgroup 1 p These are the permutations that have only 2 cycles There are the 6 transpositions with one 2 cycle green background And 3 permutations with two 2 cycles white background bold numbers 1 element Edit The trivial subgroup is the unique subgroup of order 1 Other examples EditThe even integers form a subgroup 2Z of the integer ring Z the sum of two even integers is even and the negative of an even integer is even An ideal in a ring R displaystyle R nbsp is a subgroup of the additive group of R displaystyle R nbsp A 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 See also EditCartan subgroup Fitting subgroup Fixed point subgroup Fully normalized subgroup Stable subgroupNotes Edit Gallian 2013 p 61 Hungerford 1974 p 32 Artin 2011 p 43 a b Kurzweil amp Stellmacher 1998 p 4 Jacobson 2009 p 41 Ash 2002 See a didactic proof in this video Dummit amp Foote 2004 p 90 Gallian 2013 p 81 References EditJacobson Nathan 2009 Basic algebra vol 1 2nd ed Dover ISBN 978 0 486 47189 1 Hungerford Thomas 1974 Algebra 1st ed Springer Verlag ISBN 9780387905181 Artin Michael 2011 Algebra 2nd ed Prentice Hall ISBN 9780132413770 Dummit David S Foote Richard M 2004 Abstract algebra 3rd ed Hoboken NJ Wiley ISBN 9780471452348 OCLC 248917264 Gallian Joseph A 2013 Contemporary abstract algebra 8th ed Boston MA Brooks Cole Cengage Learning ISBN 978 1 133 59970 8 OCLC 807255720 Kurzweil Hans Stellmacher Bernd 1998 Theorie der endlichen Gruppen Springer Lehrbuch doi 10 1007 978 3 642 58816 7 Ash Robert B 2002 Abstract Algebra The Basic Graduate Year Department of Mathematics University of Illinois Retrieved from https en wikipedia org w index php title Subgroup amp oldid 1180019770, wikipedia, wiki, book, books, library,
