Algebra Qual 2024-06

Let $G$ be a group, $m\in {\bf N}$, and $g\in G$ be an element such that $g^m=e$. Prove that $\operatorname{o}(g)\mid m$, where $\operatorname{o}(g)$ is the order of $g$.

Let $S_n$ denote the symmetric group on $n$ letters.
\begin{enumerate}[label=\alph*)]
	\item Is the element $(1\,2\,3\,4)(2\,5\,3\,4\,6)(1\,5\,3\,2\,4\,7)\in S_7$ even or odd? Indicate your reasoning.
	\item Find the order of $(1\,3\,4)(2\,4\,3)(1\,3\,4)\in S_4$. Show all work.
	\item Write $(1\,5\,2\,3)(2\,1\,3\,4)(1\,5\,2\,3)^{-1}\in S_5$ in disjoint cycle form. Show all work.
\end{enumerate}

Let $R$ be a commutative ring with $1$, and $N$ the ideal
\[
	N=\{a\in R\,\mid\, a^n=0\text{ for some }n\}.
\]
Let $[b]$ be the image of $b\in R$ in $R/N$. Prove that if $[a]\in R/N$ and $[a]^m=0$ then $[a]=[0]$.

Let ${\bf Z}[i]=\{a+bi\,\mid \, a,b\in {\bf Z},\, i^2=-1\}$, a subring of ${\bf C}$. Prove there is no ring homomorphism ${\bf Z}[i]\to {\bf Z}_{19}$, but there is a ring homomorphism ${\bf Z}[i]\to {\bf Z}_{13}$. Note a ring homomorphism of commutative rings with $1$ must send $1$ to $1$.

\medskip
\noindent {\itshape Hint:} The group of units in ${\bf Z}_{19}$ is the cyclic group $U(19)$ of order 18, and the group of units in ${\bf Z}_{13}$ is the cyclic group $U(13)$ of order 12.

Let $T:{\bf R}^4\to {\bf R}^4$ be orthogonal projection to the $2$-dimensional plane $P$ spanned by the vectors ${\bf v}=(2,0,1,0)$ and ${\bf w}=(-1,0,2,0)$.
\begin{enumerate}[label=\alph*)]
	\item Find (with proof) all eigenvalues and eigenvectors, along with their geometric and algebraic multiplicities.
	\item Find the matrix representing $T$ with respect to the standard basis. Is this matrix diagonalizable? Why or why not?
\end{enumerate}