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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 usually 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.

A **proper subgroup** of a group *G* is a subgroup *H* which is a proper subset of *G* (that is, *H* ≠ *G*). This is usually 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. The group *G* is sometimes denoted by the ordered pair (*G*, ∗), usually to emphasize the operation ∗ when *G* carries multiple algebraic or other structures.

## Basic properties of subgroups

- A subset
*H*of the group*G*is a subgroup of*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 if and only if*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 an inclusion homomorphism (that is, 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.^{[3]}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. Another example is the union of the x-axis and the y-axis in the plane (with the addition operation); each of these objects is a subgroup but their union is not. This also serves as an example of two subgroups, whose intersection is precisely the identity. - 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 containing*S*; 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, 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.

## 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 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*|.^{[4]}^{[5]}

**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 Z_{8}

Let *G* be the cyclic group Z_{8} whose elements are

and whose group operation is addition modulo eight. 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 S_{4 }(the symmetric group on 4 elements)

Every group has as many small subgroups as neutral elements on the main diagonal:

The trivial group and two-element groups Z_{2}. These small subgroups are not counted in the following list.

### 12 elements

### 8 elements

### 6 elements

### 4 elements

### 3 elements

## Other examples

- The even integers are a subgroup of the additive group of integers: when you add two even numbers, you get an even number.
- 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.
- Let be an abelian group; the elements of that have finite period form a subgroup of called the torsion subgroup of .

## See also

## Notes

## References

- Jacobson, Nathan (2009),
*Basic algebra*,**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.