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

Particular classes of subgroups include:

• Characteristic subgroup
• Essential subgroup
• Normal subgroup