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: These correspond to the conditions on a group, with the exception that the associative property is necessarily inherited.
 * 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, $$x, y \in S \Rightarrow xy \in S$$.

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