Algebraic structure → Group theoryGroup theory |
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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* < *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 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*a*^{n−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.

- The identity of a subgroup is the identity of the group: if G is a group with identity e
_{G}, and H is a subgroup of G with identity e_{H}, then*e*=_{H}*e*._{G} - 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*, then_{H}*ab*=*ba*=*e*._{G} - 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 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: is not a subgroup of 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 is not a subgroup of - 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 (the integers mod*n*) for some positive integer n, then n is the smallest positive integer for which*a*=^{n}*e*, and n is called the*order*of a. If ⟨*a*⟩ is isomorphic to 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.

Main 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 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*_{1} ~ *a*_{2} if and only if 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|.^{[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.

Let G be the cyclic group Z_{8} 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.^{[9]}

S_{4} 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.

Like each group, S_{4} is a subgroup of itself.

The alternating group contains only the even permutations.

It is one of the two nontrivial proper normal subgroups of S_{4}. (The other one is its Klein subgroup.)

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)

The trivial subgroup is the unique subgroup of order 1.

- The even integers form a subgroup of the integer ring the sum of two even integers is even, and the negative of an even integer is even.
- An ideal in a ring R is a subgroup of the additive group of R.
- 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.