Subgroup

Let $(G,\ast)$ be a group, with identity element $e_G$ and inverses $a^{-1}$ as provided by Uniqueness of the Identity Element and of Inverses in a Group. A subset $H\subseteq G$ is called a subgroup of $(G,\ast)$ if the following three conditions hold.

1. $e_G\in H$.
2. $a\ast b\in H$ for all $a,b\in H$.
3. $a^{-1}\in H$ for all $a\in H$.

We write $H\le G$ to indicate that $H$ is a subgroup of $(G,\ast)$. Condition 2 is the statement that $H$ is closed under $\ast$ in the sense of Binary Operation on a Set, so a subgroup carries the restriction of $\ast$ to $H$ as a binary operation.