Algebra Qual 2016-09

Problem 1

Suppose $A$ and $B$ are subgroups of a group $G$, and suppose $B$ is of finite index in $G$.
(a) Show that the index of $A\cap B\le A$ is finite, and in fact $|A:A\cap B|\le |G:B|$. Hint: Find a set map $A/A\cap B\to G/B$.
(b) Prove that equality holds in (a) if and only if $G=AB$.

Problem 2

Let $G$ be a group. Prove that $G$ is non-cyclic if and only if $G$ is the union of its proper subgroups.

Problem 3

(a) Write down an irreducible cubic polynomial over ${\mathbf{F}}_2$.
(b) Construct a field with exactly eight elements and write down its multiplication table.

Problem 4

Consider the following matrix:

(a) Determine the characteristic and minimal polynomials of $A$.
(b) Find a basis for ${\mathbf{R}}^4$ consisting of generalized eigenvectors of $A$.
(c) Find an invertible matrix $S$ such that $S^{-1}AS$ is in Jordan canonical form.
(d) Determine a Jordan canonical form of $A$.

Problem 5

Let $V$ be a vector space and $T:V\to V$ be a linear transformation.
(a) Prove that if $T$ is a projection (i.e., $T^2=T$), then $V$ can be decomposed into the internal direct sum $V=\mathrm{null}(T)\oplus \mathrm{range}(T)$.
(b) Suppose $V$ is an inner product space and $T^{\ast }$ is the adjoint of $T$ with respect to the inner product. Show that $\mathrm{null}(T^{\ast })$ is the orthogonal complement of $\mathrm{range}(T)$.
(c) Suppose $V$ is an inner product space and $T$ is an orthogonal projection, i.e., a projection for which the null space and range are orthogonal. Show that $T$ is self adjoint.