Algebra Qual 2018-06

Problem 1

Let $G$ be a group and $a \in G$ be an element. Let $n \in N$ be the smallest positive number such that $a^{n} = e$ , where $e$ is the identity element. Show that the set

{e, a, a^{2}, \dots, a^{n - 1}}

contains no repetitions.

Problem 2

Let $G$ be a finite group and $H, K ⊴ G$ be normal subgroups of relatively prime order. Prove that $G$ is isomorphic to a subgroup of $G / H \times G / K$ .

Problem 3

Prove that if $ϕ : R \to S$ is a surjective ring homomorphism between commutative rings with unity, then $ϕ (1_{R}) = 1_{S}$ .

Problem 4

Let $V \subset R^{5}$ be the subspace defined by the equation

x_{1} - 2 x_{2} + 3 x_{3} - 4 x_{4} + 5 x_{5} = 0.

(a) Find (with justification) a basis for $V$ .
(b) Find (with justification) a basis for $V^{⊥}$ , the subspace of $R^{5}$ orthogonal to $V$ under the usual dot product.

Problem 5

Suppose $V$ is a finite-dimensional real vector space and $T : V \to V$ is a linear transformation. Prove that $T$ has at most $\dim (range T)$ distinct nonzero eigenvalues.