Algebra Qual 2015-03

Let $G$ and $H$ be groups of order 10 and 15, respectively. Prove that if there is a nontrivial homomorphism $\phi:G\to H$, then $G$ is abelian.

Let $G$ be an abelian group and $G_T$ be the set of elements of finite order in $G$.

\medskip
\begin{enumerate}[label=(\alph*)]
	\item Prove that $G_T$ is a subgroup of $G$.
	\item Prove that every non-identity element of $G/G_T$ has infinite order.
	\item Characterize the elements of $G_T$ when $G={\bf R}/{\bf Z}$, where ${\bf R}$ is the additive group of real numbers.
\end{enumerate}

\begin{enumerate}
	\item Suppose $I$ and $J$ are ideals in a commutative ring $R$ such that $R=I+J$. Prove that the map $f:R\to R/I\times R/J$ given by $f(x)=(x+I,x+J)$ induces the isomorphism
	\[
		R/IJ\cong R/I\times R/J.
	\]
	\item Prove that $\left({\bf Z}/3{\bf Z}\right)[x]/(x^3-x^2-1)\cong \left({\bf Z}/3{\bf Z}\right)[x]/(x^3+x+1)$. ({\itshape Hint:} Use part (a).)
\end{enumerate}

An element $r$ of a ring $R$ is said to be {\bfseries idempotent} if $r^2=r$. Suppose that $R$ is a commutative ring with unity containing an idempotent element $e$.

\medskip
\begin{enumerate}[label=(\alph*)]
	\item Prove that $1-e$ is also idempotent.
	\item Prove that $eR$ and $(1-e)R$ are both ideals in $R$ and that
	\[
		R\cong eR\times (1-e)R.
	\]
	\item Prove that if $R$ has a unique maximal ideal, then the only idempotent elements in $R$ are 0 and 1.
\end{enumerate}

Suppose $T$ is a linear transformation on a finite-dimensional complex inner-product space $V$. Let $I$ denote the identity transformation on $V$. The {\bfseries numerical range} of $T$ is the subset of ${\bf C}$ defined by
\[
	\operatorname{W}(T)=\{\langle T(x),x\rangle \,|\, x\in V,\quad \|x\|=1\}.
\]

\medskip
\begin{enumerate}[label=(\alph*)]
	\item Show that $\operatorname{W}(T+cI)=\operatorname{W}(T)+c$ for every $c\in {\bf C}$.
	\item Show that $\operatorname{W}(cT)=c\operatorname{W}(T)$ for every $c\in {\bf C}$.
	\item Show that the eigenvalues of $T$ are contained in $\operatorname{W}(T)$.
	\item Let $\mathcal{B}$ be an orthonormal basis for $V$. Show that the diagonal entries of $[T]_{\mathcal{B}}$ are contained in $\operatorname{W}(T)$.
\end{enumerate}