Subgroup

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 $x,y\in S\Rightarrow xy^{-1}\in S$ .

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 on a given group include:

Centre of a group [r]: The subgroup of a group consisting of all elements which commute with every element of the group. ^[e]
Commutator subgroup [r]: The subgroup of a group generated by all commutators. ^[e]
Frattini subgroup [r]: The intersection of all maximal subgroups of a group. ^[e]

References