\documentclass[a4paper, 10pt]{article}
\usepackage[estonian]{babel}
\usepackage{t1enc}
\usepackage{amsmath}
\usepackage{amssymb}
\usepackage{rotating}

\newcommand{\Z}{\mathbb{Z}}
\newcommand{\N}{\mathbb{N}}
\newcommand{\F}{\mathbb{F}}
\newcommand{\Q}{\mathbb{Q}}
\newcommand{\R}{\mathbb{R}}
\newcommand{\C}{\mathbb{C}}
\renewcommand{\P}{\mathbb{P}}

\topmargin-8em
\oddsidemargin0cm
\textwidth16cm
\textheight32cm
\evensidemargin-2cm

\begin{document}

\pagestyle{empty}

\Large

{\hfill{Fall 2021} \smallskip %
\vspace{2em}

\begin{center}
{\LARGE Introduction to Algebraic Structures\\} 

{\LARGE Problem Set 2\\}
\vspace{1em}
{\LARGE Group theory.}
\end{center}
}



\bigskip\smallskip

{\fontsize{10}{12}\selectfont Cauchy's Theorem: If $p\in\P$ and $p\mid|G|$ for some \emph{finite} group $G$, then $G$ has an element of order $p$.}

\bigskip

\noindent 1. Consider the groups $(\Z^5,+)$ and $(\Z^2,+)$, and let $$f(a,b,c,d,e) = (a+2b+3c,4b+5d+6e).$$ Prove that $f:\Z^5\rightarrow \Z^2$ is a homomorphism and find the kernel $\mathrm{Ker}(f)$, up to isomorphism. Is $f$ surjective?

\bigskip

\noindent 2. Find all group homomorphisms from ($\Z_4$,+) to ($S_4$,$\circ$).

\bigskip

\noindent 3. Let $p$ be a prime and $G$ a finite group. Prove that if the order of every element of $G$ is $p^l$, $l\in\N_0$, then $|G|=p^k$, $k\in\N_0$ (i.e. $G$ is a $p$-group). 

\bigskip

\noindent 4. Let $G$ be a \emph{finite} group and $f$ an endomorphism of $G$ such that $f^2$ is the identity map and $f(g)=g$ implies $g=1$. Prove that $G$ is Abelian.
\\\hfill\vskip-30pt\begin{turn}{180}{\tiny{\it Hint.} Show that $f(x)=x^{-1}$ for all $x = gf(g)^{-1}$, and that all $x$ are of that form.}\hfill\end{turn}

\bigskip

\noindent 5. Let $G_1\leq G_2\leq\ldots\leq G_k\leq \ldots$ be simple groups. Prove that $\bigcup\limits_{k\in\N}G_k$ is also a simple \emph{group}.

\bigskip

\noindent 6. Suppose $G$ is a finite group, $H$ a normal subgroup of $G$ and the order of \emph{some} element $gH \in G/H$ is 2021. Prove that $2021$ divides the order of $g$. (Extra question for no extra credit: Does the same hold if $G$ is infinite?)

\bigskip

\noindent 7. Let $H\unlhd G$, $K\unlhd G$ and $G=HK$. Prove that $G/(H\cap K)\cong \left(G/H\right)\times \left(G/K\right)$.

\bigskip

\noindent 8. Suppose $G$ is a group with normal divisor $H$, $|H|=5$ and $G/H\cong S_4$. Show that $|G|=120$ and $G$ has a normal divisor $K$ so that $|K|=20$.

\bigskip

\noindent 9${^*}$. Let $p$ and $q$ be different primes. Prove that the number of nonisomorphic non-Abelian groups of order $pq$ is either 0 or 1. Use of Cauchy's Theorem is allowed. 

\bigskip

\noindent 10${^{\circ}}$. Let $p\in\P$. 

* \; Prove that there are \emph{no more} than 5 groups of order $p^3$. 

** \hskip2pt Prove that there are \emph{exactly} 5 groups of order $p^3$. 

\end{document}