Algebra Qual 2022-06

Problem 1

Let $N$ be a finite normal subgroup of $G$. Prove there is a normal subgroup $M$ of $G$ such that $[G:M]$ is finite and $nm=mn$ for all $n\in N$ and $m\in M$.

Hint: You may use the fact that the centralizer $C(h):=\{g\in G\mid gh{g}^{-1}=h\}$ is a subgroup of $G$.)

Problem 2

Let $S_7$ denote the symmetric group.
(a) Give an example of two non-conjugate elements of $S_7$ that have the same order.
(b) If $g\in S_7$ has maximal order, what is the order of $g$?
(c) Does the element $g$ that you found in part (b) lie in $A_7$? Fully justify your answer.
(d) Determine whether the set $\{h\in S_7\mid |h|=|g|\}$ is a single conjugacy class in $S_7$, where $g$ is the element you found in part (b).

Problem 3

Let $R$ be a commutative ring with $1$. Use theorems in ring theory to prove:
(a) $⟨x⟩$ is a prime ideal in $R[x]$ if and only if $R$ is an integral domain.
(b) $⟨x⟩$ is a maximal ideal in $R[x]$ if and only if $R$ is a field.

Problem 4

Let $R$ be a commutative ring with $1$, and $\sigma :R\to R$ be a ring automorphism.
(a) Show that $F=\{r\in R\mid \sigma (r)=r\}$ is a subring of $R$ (with $1$).
(b) Show that if ${\sigma }^2$ is the identity map on $R$, then each element of $R$ is the root of a monic polynomial of degree 2 in $F[x]$, where $F$ is as in (a).

Problem 5

Let $A=\left[\begin{array}{ccc}2& -1& -1\\ 1& 0& -1\\ 1& -1& 0\end{array}\right]$.

(a) Compute the characteristic polynomial $p_A(x)$ of $A$. It has integer roots.
(b) For each eigenvalue $\lambda$ of $A$, find a basis for the eigenspace $E_{\lambda }$.
(c) Determine if $A$ is diagonalizable. If so, give matrices $P$ and $B$ such that $P^{-1}AP=B$ and $B$ is diagonal. If no, explain carefully why $A$ is not diagonalizable.