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Formal Definition
If a subset $H$ of a group $G$ is closed under the binary operation of $G$ and if $H$ with the induced operation from $G$ is itself a group, then $H$ is a subgroup of $G$. We shall let $H \le G$ or $G \ge H$ denote that $H$ is a subgroup of $G$, and $H<G$ or $G>H$ shall mean $H\le G$ but $H\neq G$.
Informal Definition
Replace this text with an informal definition.
Example(s)
$\langle \mathbb{Q^+},\cdot \rangle$ is subgroup of $\langle \mathbb{R^+},\cdot \rangle$
Non-example(s)
$\langle\mathbb{Q^+},\cdot \rangle$ is not a subgroup of $\langle\mathbb{R},+\rangle$ because $\langle\mathbb{Q^+},\cdot\rangle$ has a different binary operation than $\langle\mathbb{R},+\rangle$.
$\langle \mathbb{Z}^{*}, + \rangle$ is not a subgroup of $\langle \mathbb{Z}, + \rangle$, since $\langle \mathbb{Z}^{*}, + \rangle$ does not contain the identity, and therefor cannot be a group.
Additional Comments
Every group $G$ has subgroups $G$ itself (called the improper subgroup of $G$ and $\{ e \}$ (called the trivial subgroup of $G$).