Subgroup/Examples
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Examples of Subgroups
$\N$ in $\struct {\R_{\ne 0}, \times}$
Consider the multiplicative group of real numbers $\struct {\R_{\ne 0}, \times}$.
Consider the algebraic structure $\struct {\N_{> 0}, \times}$ formed by the non-zero natural numbers under multiplication.
Then $\struct {\N_{> 0}, \times}$ is not a subgroup of $\struct {\R_{\ne 0}, \times}$.
Matrices $\begin{bmatrix} 1 & a \cr 0 & 1 \end{bmatrix}$ in General Linear Group
Let $\GL 2$ denote the general linear group of order $2$.
Let $H$ be the set of square matrices of the form $\begin{bmatrix} 1 & a \cr 0 & 1 \end{bmatrix}$ for $a \in \R$.
Then $\struct {H, \times}$ is a subgroup of $\GL 2$, where $\times$ is used to denote (conventional) matrix multiplication.