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*_{H}, then*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**R**^{2}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: 2**Z**∪ 3**Z**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**R**^{2}is not a subgroup of**R**^{2}. - 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**/*n***Z**(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**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.

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 *a*_{1}^{−1}*a*_{2} 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]}

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

The whole group S_{4} is a subgroup of S_{4}, of order 24. Its Cayley table is

Each element s of order 2 in S_{4} 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).

The trivial subgroup is the unique subgroup of order 1 in S_{4}.

- The even integers form a subgroup 2
**Z**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.