# Dictionary Definition

subgroup

### Noun

1 a distinct and often subordinate group within a
group

2 (mathematics) a subset (that is not empty) of a
mathematical group

# User Contributed Dictionary

## English

### Noun

- A group within a larger group; a group whose members are some, but not all, of the members of a larger group.
- A subset H of a group G that is itself a group and has the same binary operation as G.

#### Synonyms

- (group within a group): subset

#### Derived terms

#### Translations

group within a larger group

- Croatian: podgrupa
- Swedish: undergrupp

in group theory

- Croatian: podgrupa
- French: sous-groupe
- German: Untergruppe
- Italian: sottogruppo
- Portuguese: subgrupo
- Spanish: subgrupo
- Swedish: delgrupp

# Extensive Definition

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. This is
usually represented notationally by H ≤ G, read as "H is a subgroup
of G".

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 consisting of just the identity element. 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.

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 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 = an − 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 (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 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.
- 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 containing S; it is denoted by and is said to be the subgroup generated by S. An element of G is in 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 . If is isomorphic to Z/nZ 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 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 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''=

## Cosets and Lagrange's theorem

Given a subgroup H and some a in G, we define the
left coset aH = . 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 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,

- [ G : H ] =

Right cosets are defined analogously: Ha = . 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.

subgroup in Czech: Podgrupa

subgroup in Danish: Undergruppe

subgroup in German: Untergruppe

subgroup in Spanish: Subgrupo

subgroup in French: Sous-groupe

subgroup in Korean: 부분군

subgroup in Croatian: Podgrupa

subgroup in Italian: Sottogruppo

subgroup in Dutch: Ondergroep (wiskunde)

subgroup in Polish: Podgrupa

subgroup in Portuguese: Subgrupo

subgroup in Russian: Подгруппа

subgroup in Serbian: Подгрупа (математика)

subgroup in Finnish: Aliryhmä

subgroup in Vietnamese: Nhóm con

subgroup in Turkish: Altöbek

subgroup in Chinese: 子群

# Synonyms, Antonyms and Related Words

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