Subgroups

Let $G$ be a group. A subset $H$ is a subgroup of $G$ if it is a group with respect to the same binary operation. We sometimes write $H\subseteq G$ to denote that $H$ is a subgroup of $G$

Theorem: Let $G$ be a group and let $H\subseteq G$. Then $H$ is a subgroup if and only if

• $1_G\in H$
• $h_1\circ h_2\in H$ for all $h_1,h_2\in H$
• $h^{-1}\in H$ for all $h\in H$