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.

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L A T E X

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Let $G$ be a group and $a\in G$ be an element. Let $n\in {\bf N}$ be the smallest positive number such that $a^n=e$, where $e$ is the identity element. Show that the set
\[
	\{e,a,a^2,\ldots, a^{n-1}\}
\]
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 $G / H \times G / K$ .

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Let $G$ be a finite group and $H,K\mathrel{\unlhd}G$ 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 $ϕ (1_{R}) = 1_{S}$ .

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Prove that if $\phi:R\to S$ is a surjective ring homomorphism between commutative rings with unity, then $\phi(1_R)=1_S$.

Problem 4

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

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

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

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Let $V\subset {\bf R}^5$ be the subspace defined by the equation
\[
	x_1-2x_2+3x_3-4x_4+5x_5=0.
\]
\begin{enumerate}[label=\alph*)]
	\item Find (with justification) a basis for $V$.
	\item Find (with justification) a basis for $V^{\perp}$, the subspace of ${\bf R}^5$ orthogonal to $V$ under the usual dot product.
\end{enumerate}

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.

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