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Short description: Subset of a group that forms a group itself

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.[1]

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}).[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.

Subgroup tests

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 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

  • 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.[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 [math]\displaystyle{ \mathbb{Z}/8\mathbb{Z} }[/math], the integers mod 8 under addition. The subgroup H contains only 0 and 4, and is isomorphic to [math]\displaystyle{ \mathbb{Z}/2\mathbb{Z} }[/math]. 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

Main pages: 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 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,

[math]\displaystyle{ [ G : H ] = { |G| \over |H| } }[/math]

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

Let G be the cyclic group Z8 whose elements are

[math]\displaystyle{ G = \left\{0, 4, 2, 6, 1, 5, 3, 7\right\} }[/math]

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.[9]

Example: Subgroups of S4

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 elements

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

12 elements

The alternating group A4 showing only the even permutations

60px60px 60px 60px

8 elements

Dihedral group of order 8

  233px|Dihedral group of order 8

  233px|Dihedral group of order 8


6 elements

Symmetric group S3

187px|Symmetric group S3

Subgroup:Cyclic group 3; Cayley table; subgroup of S4 (elements 0,11,19).svg
187px|Symmetric group S3

Subgroup:Cyclic group 3; Cayley table; subgroup of S4 (elements 0,15,20).svg
187px|Symmetric group S3

Subgroup:Cyclic group 3; Cayley table; subgroup of S4 (elements 0,8,12).svg

4 elements

Klein four-group
142px|Klein four-group 142px|Klein four-group
Klein four-group
Cyclic group Z4
142px|Cyclic group Z4
Cyclic group Z4

3 elements

Cyclic group Z3
120px|Cyclic group Z3
Cyclic group Z3
Cyclic group Z3

2 elements

Each element s of order 2 in S4 generates a subgroup {1,s} of order 2. There are 9 such elements: the [math]\displaystyle{ \binom{4}{2} = 6 }[/math] transpositions (2-cycles) and the three elements (12)(34), (13)(24), (14)(23).

1 element

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

Other examples

  • 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 [math]\displaystyle{ R }[/math] is a subgroup of the additive group of [math]\displaystyle{ R }[/math].
  • 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