Algebra Qual 2015-03

Problem 1

Let $G$ and $H$ be groups of order 10 and 15, respectively. Prove that if there is a nontrivial homomorphism $ϕ : G \to H$ , then $G$ is abelian.

Problem 2

Let $G$ be an abelian group and $G_{T}$ be the set of elements of finite order in $G$ .
(a) Prove that $G_{T}$ is a subgroup of $G$ .
(b) Prove that every non-identity element of $G / G_{T}$ has infinite order.
(c) Characterize the elements of $G_{T}$ when $G = R / Z$ , where $R$ is the additive group of real numbers.

Problem 3

(a) Suppose $I$ and $J$ are ideals in a commutative ring $R$ such that $R = I + J$ . Prove that the map $f : R \to R / I \times R / J$ given by $f (x) = (x + I, x + J)$ induces the isomorphism

R / I J ≅ R / I \times R / J .

(b) Prove that $(Z / 3 Z) [x] / (x^{3} - x^{2} - 1) ≅ (Z / 3 Z) [x] / (x^{3} + x + 1)$ . (Hint: Use part (a).)

Problem 4

An element $r$ of a ring $R$ is said to be idempotent if $r^{2} = r$ . Suppose that $R$ is a commutative ring with unity containing an idempotent element $e$ .
(a) Prove that $1 - e$ is also idempotent.
(b) Prove that $e R$ and $(1 - e) R$ are both ideals in $R$ and that

R ≅ e R \times (1 - e) R .

Problem 5

Suppose $T$ is a linear transformation on a finite-dimensional complex inner-product space $V$ . Let $I$ denote the identity transformation on $V$ . The numerical range of $T$ is the subset of $C$ defined by

W (T) = {⟨ T (x), x ⟩ | x \in V, ∥ x ∥ = 1} .

(a) Show that $W (T + c I) = W (T) + c$ for every $c \in C$ .
(b) Show that $W (c T) = c W (T)$ for every $c \in C$ .
(c) Show that the eigenvalues of $T$ are contained in $W (T)$ .
(d) Let $B$ be an orthonormal basis for $V$ . Show that the diagonal entries of $[T]_{B}$ are contained in $W (T)$ .

Important note

Sometime after this exam was given, the exam syllabus was updated and the topic of general inner-product spaces was removed. As such, Problem 5 does not appear in the problem bank.