
\documentclass{amsart}

\title{Math studies algebra Spring 2018: Homework 1}

\author{JC}

\begin{document}

\maketitle

   This homework is due by class time on Monday 29 January. It must be typeset
   (preferably in\ \LaTeX) and submitted as a PDF file on the Canvas site,
   with a filename of the form \begin{verbatim} andrewID_alg_homeworknumber.pdf \end{verbatim}   
   For each minute that it is late, the grade will be reduced by 10 percent. 

\medskip

\begin{enumerate}

\item  Let $G$ be a finite group, let $k$ be a field and let $R$ be the set of all functions
    from $G$ to $k$. In a suggestive notation we will write a typical element $f \in R$ in the form
    $\sum_{i=1}^n a_i g_i$, where $g_1, \ldots g_n$ enumerates $G$ and $f(g_i) = a_i$.

    We define operations  $+$ and $\times$ on $R$ as follows:

\begin{itemize}

\item  $\sum_{i=1}^n a_i g_i + \sum_{i=1}^n b_i g_i = \sum_{i=1}^n c_i g_i$, where $c_i = a_i + b_i$.
   

\item  $\sum_{i=1}^n a_i g_i \times \sum_{j=1}^n b_j g_j = \sum_{k=1}^n c_k g_k$, where 
   $c_k = \sum \{ a_i b_j: g_i g_j = g_k \}$. 

\end{itemize} 

\begin{enumerate}

\item Prove that $R$ forms a ring with $1$, given these definitions of $+$ and $\times$.

\item Prove that $(R, \times)$ contains an isomorphic copy of $G$.

\item Prove that $R$ contains an isomorphic copy of $k$. 

\item Identify (with proof) the class of groups $G$ for which $R$ is commutative.

\end{enumerate}

\item  Let $k$ be a field, $G$ a finite group and $V$ a vector space over $k$ (IE a $k$-module). Suppose that $G$ acts on
    $V$ in such a way that the map $v \mapsto g \cdot v$ is linear for each $g \in G$.
    Let $R$ be the ring constructed from $k$ and $G$ in the preceding question.  

\begin{enumerate} 

\item Prove that if we define $(\sum_{i=1}^n a_i g_i) v = \sum_{i=1}^n a_i (g_i \cdot v)$, then
    $V$ becomes a left $R$-module.      

\item Prove that all left $R$-modules arise in this way.

\end{enumerate} 


\item Consider the ring $R$ of $N \times N$ real matrices. We will generalise some familiar ideas from calculus to
    this setting:

\begin{itemize} 

\item  If $(M_n)_{n in {\mathbb N}}$ is a sequence of elements of $R$, then $(M_n)$ is {\em convergent} if and only if for each
    $i$ and $j$ the sequence formed by the $(i, j)$-entries in $M_n$ is convergent. In this case the {\em limit} is the
    matrix $M$ whose $(i, j)$-entry is the limit of the sequence formed by the $(i, j)$-entries in $M_n$.

\item Similarly, an infinite sum $\sum_{n=0}^\infty M_n$ of elements of $R$ is convergent iff the sequence of partial
    sums  $\sum_{n=0}^j M_n$ is convergent, and in this case $\sum_{n=0}^\infty M_n$ is defined as the limit of the sequence of partial sums.

\end{itemize}


\begin{enumerate}


\item    Prove that for any matrix $A \in R$, the power series $\sum_{n=0}^\infty \frac{A^n}{n!}$ converges.
    
    Hint: Let $K$ be the maximum of the absolute values of the entries in $A$. Find an upper bound for the
    absolute values of the entries in $A^n$ in terms of $K$, $N$ and $n$. Now use the Comparison Test. 

\item    Compute the sum of the series $\sum_{n=0}^\infty \frac{A^n}{n!}$ when $N = 2$ and
$A = \left (
       \begin{array}{rr}
       \lambda & 0 \\
       1       & \lambda \\
       \end{array}
       \right )
$



 \item  (Challenging, not for credit) We define $\exp(A) = \sum_{n=0}^\infty \frac{A^n}{n!}$. Prove that 
   if $A B = B A$ then $\exp(A + B) = \exp (A) \exp (B)$. 

\end{enumerate}

\item   Review the following basic definitions from category theory: {\em category}, {\em object}, {\em morphism}, {\em initial object}, {\em terminal object}.

\item   Let $\mathcal C$ be a category, let $I$ be some (potentially infinite) set $I$ and let $(a_i)_{i \in I}$ be an $I$-indexed family of objects of $\mathcal C$.
     A {\em cone} over this family consists of an object $b$ and an $I$-indexed family of morphisms $(f_i : b \rightarrow a_i)_{i \in I}$ (I usually imagine
     $(a_i)$ as a horizontal line of objects and $b$ sitting above them all with arrows going down into each $a_i$)
 

   We will make a new category $\mathcal D$ as follows: The objects of $\mathcal D$ are cones over $(a_i)$. If $B$ is a cone consisting of 
   an object $b$ and morphisms $(f_i : b \rightarrow a_i)_{i \in I}$, and $C$ consists of         
   an object $c$ and morphisms $(g_i : c \rightarrow a_i)_{i \in I}$, then a morphism from $B$ to $C$ in $\mathcal D$ is a morphism
   $h$ from $b$ to $c$ in $\mathcal C$ such that $f_i = g_i \circ h$. 

         
   \begin{enumerate}

   \item   Verify that $\mathcal D$ is a category. 

   \item   A {\em product} for the sequence $(a_i)_{i \in I}$ is a terminal object in the category $\mathcal D$. Prove that
   if $\mathcal C$ is the category of left $R$-modules over a ring $R$ then every sequence has a product. 

   \end{enumerate} 




\item Let $G$ be an abelian group. A {\em torsion element} of $G$ is an element $g \in G$ such that $n g = 0$ for some
    integer $n > 0$, and $G$ is {\em torsion-free} if $0$ is the only torsion element.

\begin{enumerate}

\item  Let $G$ be any abelian group. Prove that the set $H$ of torsion elements forms a subgroup of $G$, and that  
   the quotient $G/H$ is torsion-free.

\item  Given an example of an infinite abelian group in which every element is a torsion element.


\item  Let $G$ be a torsion-free abelian group. Prove that there is a vector space $V$ over the field
    $\mathbb Q$ such that $G$ is a subgroup of $(V, +)$ and every element of $V$ has the form
    $\frac{1}{n} g$ for some integer $n > 0$ and $g \in G$ ($n$ and $g$ will not be unique here).

\end{enumerate}  


   \end{enumerate}  



\end{document} 
