# Subgroup  Main Article Discussion Related Articles  [?] Bibliography  [?] External Links  [?] Citable Version  [?] This editable Main Article is under development and subject to a disclaimer. [edit intro]

In group theory, a subgroup of a group is a subset which is itself a group with respect to the same operations.

Formally, a subset S of a group G is a subgroup if it satisfies the following conditions:

• The identity element of G is an element of S;
• S is closed under taking inverses, that is, $x\in S\Rightarrow x^{-1}\in S$ ;
• S is closed under the group operation, that is, $x,y\in S\Rightarrow xy\in S$ .

These correspond to the conditions on a group, with the exception that the associative property is necessarily inherited.

It is possible to replace these by the single closure property that S is non-empty and $x,y\in S\Rightarrow xy^{-1}\in S$ .

## Examples

The group itself and the set consisting of the identity element are always subgroups.

Particular classes of subgroups include:

• Characteristic subgroup [r]: A subgroup which is mapped to itself by any automorphism of the whole group. [e]
• Essential subgroup [r]: A subgroup of a group which has non-trivial intersection with every other non-trivial subgroup. [e]
• Normal subgroup [r]: Subgroup N of a group G where every expression g-1ng is in N for every g in G and every n in N. [e]

Specific subgroups of a given group include:

## Properties

The intersection of any family of subgroups is again a subgroup. We can therefore define the subgroup generated by a subset S of a group G, denoted $\langle S\rangle$ , to be the intersection of all subgroups of G containing S. The union of two subgroups is not in general a subgroup (indeed, it is only a subgroup if one component of the union contains the other). Instead, we may define the join of two subgroups to the subgroup generated by their union.

## Cosets

The left cosets of a subgroup H of a group G are the subsets of G of the form x H for a particular element x of G:

$xH=\{xh:h\in H\}.\,$ The right cosets H x are defined similarly:

$Hx=\{hx:h\in H\}.\,$ The subgroup H is itself one of its own cosets, namely that on the identity element.

The left cosets partition the group G, any two cosets $xH$ and $yH$ are either equal or disjoint. This may be proved directly, or deduced from the observation that the left cosets are the equivalence classes for the equivalence relation ${\stackrel {H}{\sim }}$ defined by

$x{\stackrel {H}{\sim }}y\Leftrightarrow x^{-1}y\in H.\,$ Similar remarks apply to the right cosets. In general the two partitions of the group defined by the left cosets and by the right cosets are not the same. A subgroup is normal if and only if the left cosets agree with the right cosets for all elements.

### Index

The index of a subgroup H of a group G, denoted $[G:H]$ is the number (if finite) of cosets of H in G. Two cosets may be put into one-to-one correspondence $xH\leftrightarrow yH$ by $xh\leftrightarrow yh$ , so if the cosets are finite then they all have the same order. We can now deduce

Lagrange's Theorem: In a finite group the order of a subgroup multiplied by its index equals the order of the group:
$\vert G\vert =\vert H\vert \cdot [g:h].\,$ In particular the order of a subgroup divides the order of the group, and the order of an element divides the orderof the group.

## Maximal subgroup

A subgroup M of G is maximal if M is not the whole of G but there is no other subgroup H strictly between M and G.