Subgroup

Definition

Given a Group $(G,\ast )$ and a subset $H$ of $G$, if $H$ forms a group under the operation $\ast {|}_{H\times H}$, then $H$ is called a Subgroup of $G$. We denote that $H$ is a Subgroup of $G$ by writing $H\le G$.

Examples

1. For a given $n\in {\mathbb{Z}}_{>0}$, $(n):=nx\mid x\in \mathbb{Z}$ is a Subgroup of $(\mathbb{Z},+)$.
2. $S{L}_2(\mathbb{R}):=A\in Ma{t}_{2\times 2}(\mathbb{R})\mid \det (A)=1$ is a Subgroup of $(G{L}_2(\mathbb{R}),\times )$.