Subgroup

A subgroup of a group $G$ is a subset $H \subseteq G$ that is itself a group under the same operation. This holds if and only if $H$ is non-empty, closed under the group operation, and closed under taking inverses.

Every group has at least two subgroups: the trivial subgroup $\{e\}$ containing only the identity, and the group $G$ itself. A subgroup that is neither of these is called a proper subgroup.

The subgroups of a group form a lattice ordered by inclusion, where the meet of two subgroups is their intersection and the join is the subgroup they generate together.