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 HG, read as "H is a subgroup of G".

The trivial subgroup of any group is the subgroup {e} consisting of just the identity element.

A proper subgroup of a group G is a subgroup H which is a proper subset of G (that is, HG). 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}).[1][2]

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.

Subgroup testsEdit

Suppose that G is a group, and H is a subset of G.

  • 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.)[3]
  • 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]

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 HG 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 AB or BA. 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.
  • Every element a of a group G generates a 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.
 
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.

Cosets and Lagrange's theoremEdit

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 : HaH 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|.[6][7]

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 Z8Edit

Let G be the cyclic group Z8 whose elements are

 

and whose group operation is addition modulo 8. Its Cayley table is

+ 0 4 2 6 1 5 3 7
0 0 4 2 6 1 5 3 7
4 4 0 6 2 5 1 7 3
2 2 6 4 0 3 7 5 1
6 6 2 0 4 7 3 1 5
1 1 5 3 7 2 6 4 0
5 5 1 7 3 6 2 0 4
3 3 7 5 1 4 0 6 2
7 7 3 1 5 0 4 2 6

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.

Example: Subgroups of S4Edit

Let S4 be the symmetric group on 4 elements. Below are all the subgroups of S4, listed according to the number of elements, in decreasing order.

24 elementsEdit

The whole group S4 is a subgroup of S4, of order 24. Its Cayley table is

 
The symmetric group S4 showing all permutations of 4 elements
All 30 subgroups
Simplified

12 elementsEdit

 
The alternating group A4 showing only the even permutations

Subgroups:
 
      

8 elementsEdit

 
 
Dihedral group of order 8

Subgroups:
   
 
 
Dihedral group of order 8

Subgroups:
   

6 elementsEdit

 
Symmetric group S3

Subgroup: 
 
Symmetric group S3

Subgroup: 
 
Symmetric group S3

Subgroup: 

4 elementsEdit

 
Klein four-group
 
Klein four-group
 
Klein four-group
 
Cyclic group Z4
 
Cyclic group Z4

3 elementsEdit

 
Cyclic group Z3
 
Cyclic group Z3
 
Cyclic group Z3

2 elementsEdit

Each element s of order 2 in S4 generates a subgroup {1,s} of order 2. There are 9 such elements: the   transpositions (2-cycles) and the three elements (12)(34), (13)(24), (14)(23).

1 elementEdit

The trivial subgroup is the unique subgroup of order 1 in S4.

Other examplesEdit

  • The 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   is a subgroup of the additive group of  .
  • 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 alsoEdit

NotesEdit

  1. ^ Hungerford (1974), p. 32
  2. ^ Artin (2011), p. 43
  3. ^ Kurzweil and Stellmacher (1998), p. 4
  4. ^ Kurzweil and Stellmacher (1998), p. 4
  5. ^ Jacobson (2009), p. 41
  6. ^ See a didactic proof in this video.
  7. ^ Dummit and Foote (2004), p. 90.

ReferencesEdit

  • Jacobson, 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.
  • Kurzweil, Hans; Stellmacher, Bernd (1998). Theorie der endlichen Gruppen. Springer-Lehrbuch. doi:10.1007/978-3-642-58816-7.