# Subgroup

In group theory, given a group *G* under a binary operation *, we say that some subset *H* of *G* is 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* is a group operation on *H*.

A **proper subgroup** of a group *G* is a subgroup *H* which is a proper subset of *G* (i.e. *H* ≠ *G*). The **trivial subgroup** of any group is the subgroup {*e*} consisting of just the identity element.

The same definitions apply more generally when *G* is an arbitrary semigroup, but this article will only deal with subgroups of groups. The group *G* is sometimes denoted by the ordered pair (*G*,*), usually to emphasize the operation * when *G* carries multiple algebraic or other structures.

In the following, we follow the usual convention of dropping * and writing the product *a***b* as simply *ab*.

## Basic properties of subgroups

*H*is a subgroup of the group*G*if and only if it is nonempty and closed under products and inverses. (The closure conditions mean the following: whenever*a*and*b*are in*H*, then*ab*and*a*^{−1}are also in*H*. These two conditions can be combined into one equivalent condition: whenever*a*and*b*are in*H*, then*ab*^{−1}is also in*H*.) In the case that*H*is finite, then*H*is a subgroup iff*H*is closed under products. (In this case, every element*a*of*H*generates a finite cyclic subgroup of*H*, and the inverse of*a*is then*a*^{−1}=*a*^{n − 1}, where*n*is the order of*a*.- The above condition can be stated in terms of a homomorphism; that is,
*H*is a subgroup of a group*G*if and only if*H*is a subset of*G*and there is a inclusion homomorphism (i.e., i(*a*) =*a*for every*a*) from*H*to*G*. - 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}. - The intersection of subgroups
*A*and*B*is again a subgroup. The union of subgroups*A*and*B*is a subgroup if and only if either*A*or*B*contains the other, since for example 2 and 3 are in the union of 2Z and 3Z but their sum 5 is not. - If
*S*is a subset of*G*, then there exists a minimum subgroup containing*S*, which can be found by taking the intersection of all of subgroups; it is denoted by <*S*> and is said to be 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. - Every element
*a*of a group*G*generates the cyclic subgroup <*a*>. If <*a*> is isomorphic to**Z**/*n***Z**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. (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.

## Example

Let *G* be the abelian group whose elements are

*G*={0,2,4,6,1,3,5,7}

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

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

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

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

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

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

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

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

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

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

This group has a pair of nontrivial subgroups: *J*={0,4} and *H*={0,2,4,6}, 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 group *G* is cyclic, and so are its subgroups. In general, subgroups of cyclic groups are also cyclic.

## Cosets and Lagrange's theorem

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 given by 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} iff *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

where o(*G*) and o(*H*) denote the orders of *G* and *H*, respectively. In particular, if *G* is finite, then the order of every subgroup of *G* (and the order of every element of *G*) must be a divisor of o(*G*).

*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.

## See also

de:Untergruppe es:Subgrupo fr:Sous-groupe it:Sottogruppo ko:부분군 pl:Podgrupa fi:Aliryhmä