I well remember a friend of mine in the math department of Sharif University of Technology back in 1990s who is now in MSU used to write his homework and research papers via LaTex but then I was more focused on humanities and didn’t need to learn this coding environment.

Given the fact that I am a first time user I was wondering that if someone could help me to locate the proper file for installation and also the basic help stuff for a beginner. Very soon, however, I learned that LaTeX is an open and free coding language. And that there is a jungle of downloadable files here and there!

As a first exercise today morning I coded via LaTeX a proof of Maschke’s Theorem which is an elegant result in modular representation of finite groups. Learning to do LaTeX was very much like my first time of riding a bicycle. It took a few hours of painful frustrations but the outcome is quite enjoyable and stylish.

\documentclass[12pt,a4paper]{article}

\usepackage{amsmath}

\usepackage{amsfonts}

\begin{document}

\title{Proof of Maschke Theorem}

\author {@VictorScenario First Exercise in \LaTeX{}}

\maketitle

Let G be a finite group, $F=\mathbb{R}$ or $F=\mathbb{C}$, and let V be an FG-module. If U is an FG-submodule of V, then there is an FG-submodule W of V such that

\[V= U \oplus W.\]

\textit{Proof:}

Choose any subspace $W_0$ of V such that $V=U \oplus W_0$.

Take a basis $(v_1, \dots , v_m)$ of U, extend it to a basis $(v_1, \dots, v_n)$ of V, and let \(W_0=sp(v_{m+1}, \dots ,v_n)\).

For $v \in V$, we have $v=u+w$ for unique vectors $u \in U$ and $w \in W_0$, we define $\phi : V \to V$ by setting $v\phi=u$.

Clearly $\phi$ is a projection of V with kernel $W_0$ and image U. Define $\vartheta:V \to V$ by

\[v\vartheta=\frac {1}{|G|} \sum_{g \in G} vg \phi g^{-1} \quad (v \in V). \]

Note that $\vartheta$ is an endomorphism of V and $Im \vartheta \subseteq U$. For $v \in V$ and $x \in G$,

\[(vx) \vartheta=\frac{1}{|G|}\sum_{g \in G}(vx)g \phi g^{-1}.\]

Suppose that $h=xg$. Thus

\[(vx) \vartheta=\frac {1}{|G|}\sum_{h \in G}vh \phi h^{-1}x

=\left(\frac {1}{|G|}\sum_{h \in G} vh \phi h^{-1}\right) x

=(v \vartheta)x.

\]

Hence $\vartheta$ is an FG-homomorphism. For $u \in U, g \in G \Rightarrow ug \in U \Rightarrow (ug) \phi = ug$. Therefore

\[u \vartheta=\frac {1}{|G|} \sum_{ g \in G} ug \phi g^{-1}=\frac {1}{|G|} \sum_{g \in G} (ug)g^{-1}=\frac {1}{|G|} \sum_{g \in G} u=u. \]

Let $v \in V$. Then $v \vartheta \in U \Rightarrow (v \vartheta) \vartheta = v \vartheta \Rightarrow \vartheta^{2}=\vartheta$. Hence $\vartheta$ is a projection.

Also $Im \vartheta = U$. Now if we let $W=Ker \vartheta$ $\Rightarrow$ W is an FG-submodule of V and $V=U \oplus W$. \

\textit{Source of the Proof: Gordon James, Martin Liebeck, (2001), Representations and Characters of Groups, Second Edition, Cambridge University Press.}

\end{document}