# Michigan-State-University-Abstract-Algebra-eye-catch

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- Condition that a Matrix is Similar to the Companion Matrix of its Characteristic Polynomial Let $A$ be an $n\times n$ complex matrix. Let $p(x)=\det(xI-A)$ be the characteristic polynomial of $A$ and write it as \[p(x)=x^n+a_{n-1}x^{n-1}+\cdots+a_1x+a_0,\] where $a_i$ are real numbers. Let $C$ be the companion matrix of the polynomial $p(x)$ given […]
- Every Group of Order 20449 is an Abelian Group Prove that every group of order $20449$ is an abelian group. Outline of the Proof Note that $20449=11^2 \cdot 13^2$. Let $G$ be a group of order $20449$. We prove by Sylow's theorem that there are a unique Sylow $11$-subgroup and a unique Sylow $13$-subgroup of […]
- The Symmetric Group is a Semi-Direct Product of the Alternating Group and a Subgroup $\langle(1,2) \rangle$ Prove that the symmetric group $S_n$, $n\geq 3$ is a semi-direct product of the alternating group $A_n$ and the subgroup $\langle(1,2) \rangle$ generated by the element $(1,2)$. Definition (Semi-Direct Product). Internal Semi-Direct-Product Recall that a group $G$ is […]
- If the Order of a Group is Even, then the Number of Elements of Order 2 is Odd Prove that if $G$ is a finite group of even order, then the number of elements of $G$ of order $2$ is odd. Proof. First observe that for $g\in G$, \[g^2=e \iff g=g^{-1},\] where $e$ is the identity element of $G$. Thus, the identity element $e$ and the […]
- Find a Nonsingular Matrix $A$ satisfying $3A=A^2+AB$ (a) Find a $3\times 3$ nonsingular matrix $A$ satisfying $3A=A^2+AB$, where \[B=\begin{bmatrix} 2 & 0 & -1 \\ 0 &2 &-1 \\ -1 & 0 & 1 \end{bmatrix}.\] (b) Find the inverse matrix of $A$. Solution (a) Find a $3\times 3$ nonsingular matrix $A$. Assume […]
- Determine Eigenvalues, Eigenvectors, Diagonalizable From a Partial Information of a Matrix Suppose the following information is known about a $3\times 3$ matrix $A$. \[A\begin{bmatrix} 1 \\ 2 \\ 1 \end{bmatrix}=6\begin{bmatrix} 1 \\ 2 \\ 1 \end{bmatrix}, \quad A\begin{bmatrix} 1 \\ -1 \\ 1 […]
- Nilpotent Element a in a Ring and Unit Element $1-ab$ Let $R$ be a commutative ring with $1 \neq 0$. An element $a\in R$ is called nilpotent if $a^n=0$ for some positive integer $n$. Then prove that if $a$ is a nilpotent element of $R$, then $1-ab$ is a unit for all $b \in R$. We give two proofs. Proof 1. Since $a$ […]
- Determine the Values of $a$ so that $W_a$ is a Subspace For what real values of $a$ is the set \[W_a = \{ f \in C(\mathbb{R}) \mid f(0) = a \}\] a subspace of the vector space $C(\mathbb{R})$ of all real-valued functions? Solution. The zero element of $C(\mathbb{R})$ is the function $\mathbf{0}$ defined by […]