Algebra Qual 2014-03

Problem 1

Let $G$ be a group and suppose $\mathrm{Aut}(G)$ is trivial.
(a) Show that $G$ is abelian.
(b) Show that for any abelian group $H$, the inversion map $\varphi (h)=h^{-1}$ is an automorphism.
(c) Use parts (a) and (b) above to show that $g^2$ is the identity element for every $g\in G$.

Problem 2

Let $G$ be the group of upper-triangular real matrices $\left[\begin{array}{cc}a& b\\ 0& d\end{array}\right]$ with $a,d\ne 0$, under matrix multiplication. Let $S$ be the subset of $G$ defined by $d=1$. Show that $S$ is normal and that $G/S\cong {\mathbf{R}}^{\times }$, the multiplicative group of nonzero real numbers.

Problem 3

Let $A$ be a commutative ring with unit. We call $A$ Boolean if $a^2=a$ for every $a\in A$. Prove that in a Boolean ring $A$ each of the following holds:
(a) $2a=0$ for every $a\in A$.
(b) If $I$ is a prime ideal then $A/I$ is a field with two elements (and in particular $I$ is maximal).
(c) If $I=(a,b)$ is the ideal generated by $a$ and $b$ then $I$ can be generated by the single element $a+b+ab$. Conclude that every finitely generated ideal is principal.

Problem 4

Let $a,b\in \mathbf{R}$ and $T:{\mathbf{R}}^3\to {\mathbf{R}}^3$ be the linear transformation which is reflection across the plane $z=ax+by$.
(a) Find the eigenvalues of $T$ and for each find a basis for the corresponding eigenspace.
(b) Is $T$ diagonalizable? Justify.
(c) What is the characteristic polynomial of $T$?
(d) What is the minimal polynomial of $T$?

Problem 5

Let $\varphi :V\to W$ be a surjective linear transformation of finite-dimensional linear spaces. Show that there is a $U\subseteq V$ such that $V=(\ker (\varphi ))\oplus U$ and $\varphi {\mid }_U:U\to W$ is an isomorphism. (Note that $V$ is not assumed to be an inner-product space; also note that $\ker (\varphi )$ is sometimes referred to as the null space of $\varphi$; finally, $\varphi {\mid }_U$ denotes the restriction of $\varphi$ to $U$.)