Let $(G,\ast)$ be a \reftext{def:group-2026a}{group}, with identity element $e_G$ and inverses $a^{-1}$ as provided by \ref{thm:group-identity-inverse-uniqueness-2026a}. A subset $H\subseteq G$ is called a \textbf{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 \ref{def:binary-operation-set-2026a}, so a subgroup carries the restriction of $\ast$ to $H$ as a binary operation.