# 1 Basic typesetting

Of course you can type paragraphs containing some mathematics, such as the following one.

If $G=\operatorname{GL}_2$ then ${G_{\operatorname{ad}}}=\operatorname{PGL}_2$ and so, as above, $G_1=\operatorname{SL}_2$. Now $G_2$ is the set of $(g,h)\in \operatorname{SL}_2\times \operatorname{GL}_2$ with $g=h$ in $\operatorname{PGL}_2$, so $h=\lambda g$ for some unique $\lambda \in \mathbf{G}_m$ and $G_2=\operatorname{SL}_2\times \mathbf{G}_m$, with the obvious map to $\operatorname{GL}_2$ sending $\mathbf{G}_m$ into the centre (or perhaps its inverse depending on how one is thinking about things, but this doesn’t matter). The subgroup $\mu _2$ is embedded diagonally of course, because it’s the kernel of the map $G_2\to G$. Finally we push out via $\mu _2\to \mathbf{G}_m$ and this gives us $\operatorname{SL}_2\times \mathbf{G}_m\times \mathbf{G}_m$ modulo the subgroup of order 2 with non-trivial element $(-1,-1,-1)$. But there’s an automorphism of $\mathbf{G}_m\times \mathbf{G}_m$ sending $(-1,-1)$ to $(-1,1)$ (namely, send $(x,y)$ to $(x,xy)$) so again ${\tilde{G}}$ is just $G\times \mathbf{G}_m$.

You can also use displayed formulas such as:

$\int _{I \times \Sigma } \Phi ^*\omega = \int _I\left(\int _{\Phi _t(\Sigma )} \iota _X \omega \right)dt.$

and refer to displayed formulas such as Equation 1 below.

\begin{equation} \label{eq:stokes} \int _M d\omega = \int _{\partial M} \omega \end{equation}
1

Commutative diagrams using tikz-cd are supported as well. 