Algebra Qual 2015-09

Problem 1

Determine the number of group homomorphisms $ϕ$ between the given groups. Here $K_{4}$ denotes the Klein four-group (also known as $Z / 2 Z \times Z / 2 Z$ ) and $S_{3}$ denotes the symmetric group on three elements.
(a) $ϕ : K_{4} \to Z / 2 Z$
(b) $ϕ : Z / 2 Z \to K_{4}$
(c) $ϕ : S_{3} \to K_{4}$
(d) $ϕ : K_{4} \to S_{3}$

Problem 2

(a) Show that if $G$ is any group (not necessarily finite) and $H$ is a subgroup, then $G$ is a disjoint union of left cosets of $H$ .
(b) State and prove Lagrange's Theorem for finite groups.

Problem 3

Let $R$ be an integral domain. Suppose that $a$ and $b$ are non-associate irreducible elements in $R$ , and the ideal $(a, b)$ generated by $a$ and $b$ is a proper ideal. Show that $R$ is not a principal ideal domain (PID).

Problem 4

Let $F$ be a field and let $α$ be an element that generates a field extension of $F$ of degree five. Prove that $α^{2}$ generates the same extension.

Problem 5

Let $A = [\begin{matrix} - 2 & 1 & - 1 \\ 5 & - 2 & 2 \\ 7 & - 3 & 3 \end{matrix}]$ .
(a) Find the characteristic polynomial and the minimal polynomial of $A$ .
(b) Find the Jordan canonical form of the matrix $A$ .

Important note

Sometime after this exam was given, the exam syllabus was updated and the topic of Jordan canonical forms was removed. As such, Problem 5 does not appear in the problem bank.