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\begin{document}

\textbf{Math 6310, Assignment 1 \hfill Due in class: Friday, September 4}

% Remove the percent sign at the beginning of a line below, and then TeX will process it.
% \medskip {\large Name: }

\begin{enumerate}

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\item Suppose $G$ is a finite set with an associative law of composition, and $e\in G$ is an element such that $xe=x=ex$ for all $x\in G$. If $G$ has the property that 
\[
xz=yz \quad \text{implies} \quad x=y,
\]
prove that $G$ is a group.

%\textbf{Solution:} Let $z$ be an element of $G$; we need to show that it has an inverse. The map $g\mapsto gz$ is injective by the property displayed above; blah, blah.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\item Let $G$ be a group, and let $H$ be a subgroup of finite index. Prove that the number of right cosets of $H$ equals the number of left cosets.

%\textbf{Solution:} We verify that the map below is bijective:
%\[
%f \colon \{ \text{left cosets} \} \to \{ \text{right cosets} \}, \quad \text{where} \quad f(xH)=Hx^{-1}.
%\]

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\item Let $a$ be an element of a group $G$. Prove that there exists $x$ in $G$ with $x^2ax=a^{-1}$ if and only if $a$ is the cube of some element of $G$.

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\item Let $G$ be a finite group. If $x^2=e$ for each $x\in G$, prove that $|G|$ is a power of $2$.

%\textbf{Solution:} Let $x,y\in G$. Then $(xy)^2 = xyxy = e$, and it follows that $yx = xy$. Thus, $G$ is abelian. We continue by induction on $|G|$.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\item Find a group with elements $a,b$ such that $a$ and $b$ have finite order, but $ab$ does not have finite order. (Hint: Try looking in $\Gl_2(\bbZ)$, the group of invertible $2\times 2$ matrices over $\bbZ$.)

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\item Let $n$ be a positive integer. Consider the set $G$ of positive integers less than or equal to $n$ that are relatively prime to $n$. The number of elements of $G$ is the \emph{Euler phi-function}, denoted $\phi(n)$.

\begin{enumerate}
\item Show that $G$ is a group under multiplication modulo $n$.
\item If $m$ and $n$ are relatively prime positive integers, show that $m^{\phi(n)} \equiv 1 \mod n$.
\end{enumerate}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\item Let $n$ be a positive integer. Show that $\displaystyle{n=\sum_{d | n} \phi(d)}$, where the sum is taken over all positive integers~$d$ that divide~$n$. (Hint: A cyclic group of order $n$ has a unique subgroup of order $d$ for each $d$ dividing~$n$.)

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\item Let $G$ be a finite group with the property that for each integer $d\ge 1$, the equation $x^d=e$ has at most~$d$ solutions in $G$. Prove that $G$ is cyclic.

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\item Let $G$ be a group such that for a fixed integer $n>1$, we have $(xy)^n=x^ny^n$ for all $x,y \in G$. Let
\[
G^{(n)}=\{x^n \mid x\in G\}\quad\text{and} \quad G_{(n)}=\{x\in G \mid x^n=e\}\,.
\]
\begin{enumerate}
\item Prove that $G^{(n)}$ and $G_{(n)}$ are normal subgroups of $G$.

\item If $G$ is finite, show that the order of $G^{(n)}$ equals the index of $G_{(n)}$.

\item Show that for all $x,y \in G$, we have $x^{1-n}y^{1-n} = (xy)^{1-n}$. Use this to get $x^{n-1}y^n = y^nx^{n-1}$.

\item Conclude that elements of $G$ of the form $x^{n(n-1)}$ generate an abelian subgroup.
\end{enumerate}

%\textbf{Solution:} The hypothesis implies that $x\mapsto x^n$ is a homomorphism. The kernel is $G_{(n)}$, which is therefore normal. The image is the subgroup $G^{(n)}$, which is also normal since ...

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\item Let $G$ be a group such that $(xy)^3=x^3y^3$ for all $x,y\in G$, and such that the map $x \mapsto x^3$ is bijective. Prove that the group $G$ is abelian.

\end{enumerate}
\end{document}

