Algebra Qual 2024-06

Problem 1

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

Problem 2

Let $S_n$ denote the symmetric group on $n$ letters.
(a) Is the element $(1234)(25346)(153247)\in S_7$ even or odd? Indicate your reasoning.
(b) Find the order of $(134)(243)(134)\in S_4$. Show all work.
(c) Write $(1523)(2134)(1523{)}^{-1}\in S_5$ in disjoint cycle form. Show all work.

Problem 3

Let $R$ be a commutative ring with $1$, and $N$ the ideal

Let $[b]$ be the image of $b\in R$ in $R/N$. Prove that if $[a]\in R/N$ and $[a{]}^m=0$ then $[a]=[0]$.

Problem 4

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

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

Problem 5

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