Subgroups

Definition

Given a group $(G, \circ)$ and $H \subset G$, suppose $(H, \circ\mid _{\mid H})$ is also group. Then we call it a subgroup of $(G, \circ)$. If $H \subsetneq G$, we call $H$ a proper subgroup.

As we sometimes refer to the group $G$ without reference to the operation $\circ$, we similarly sometimes refer to the subgroup $H$ without reference to the operation $\circ_{\mid H}$.

Examples

Trivial subgroup. Suppose $e \in G$ is the identity element of the group $(G, \circ)$. Then $\set{e}$ is a subgroup of of $(G, \circ)$.