(group-theory-basics)=
# Basics

This pages contains fundamental results of group theory.

## Theorems

:::{prf:theorem} First Isomorphism Theorem

Let $G$ and $H$ be two groups, and $f\colon G\to H$ a homomorphism.  
Let $N$ be a normal subgroup of $G$, and $\varphi\colon G\to G/N$ the natural surjective homomorphism. If $N\subseteq\ker f$, then there exists a unique homomorphism $h\colon G/N\to H$ such that the following diagram commutes:

% https://q.uiver.app/#q=WzAsMyxbMCwwLCJHIl0sWzIsMCwiSCJdLFswLDIsIkcvTiJdLFswLDEsImYiXSxbMiwxLCJoIiwyLHsic3R5bGUiOnsiYm9keSI6eyJuYW1lIjoiZGFzaGVkIn19fV0sWzAsMiwiXFx2YXJwaGkiLDJdXQ==

:::{tikz}
\begin{tikzcd}
	G && H \\
	\\
	{G/N}
	\arrow["f", from=1-1, to=1-3]
	\arrow["\varphi"', from=1-1, to=3-1]
	\arrow["h"', dashed, from=3-1, to=1-3]
\end{tikzcd}
:::

:::

:::{prf:proof}

TODO
:::