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• Interchangeability of Limits and Probability of Increasing or Decreasing Sequence of Events A sequence of events $\{E_n\}_{n \geq 1}$ is said to be increasing if it satisfies the ascending condition \[E_1 \subset E_2 \subset \cdots \subset E_n \subset \cdots.\] Also, a sequence $\{E_n\}_{n \geq 1}$ is called decreasing if it satisfies the descending condition \[E_1 […]
• If a Sylow Subgroup is Normal in a Normal Subgroup, it is a Normal Subgroup Let $G$ be a finite group. Suppose that $p$ is a prime number that divides the order of $G$. Let $N$ be a normal subgroup of $G$ and let $P$ be a $p$-Sylow subgroup of $G$. Show that if $P$ is normal in $N$, then $P$ is a normal subgroup of $G$.   Hint. It follows from […]
• A Group of Linear Functions Define the functions $f_{a,b}(x)=ax+b$, where $a, b \in \R$ and $a>0$. Show that $G:=\{ f_{a,b} \mid a, b \in \R, a>0\}$ is a group . The group operation is function composition. Steps. Check one by one the followings. The group operation on $G$ is […]
• The Quotient Ring by an Ideal of a Ring of Some Matrices is Isomorphic to $\Q$. Let \[R=\left\{\, \begin{bmatrix} a & b\\ 0& a \end{bmatrix} \quad \middle | \quad a, b\in \Q \,\right\}.\] Then the usual matrix addition and multiplication make $R$ an ring. Let \[J=\left\{\, \begin{bmatrix} 0 & b\\ 0& 0 \end{bmatrix} […]
• Trace, Determinant, and Eigenvalue (Harvard University Exam Problem) (a) A $2 \times 2$ matrix $A$ satisfies $\tr(A^2)=5$ and $\tr(A)=3$. Find $\det(A)$. (b) A $2 \times 2$ matrix has two parallel columns and $\tr(A)=5$. Find $\tr(A^2)$. (c) A $2\times 2$ matrix $A$ has $\det(A)=5$ and positive integer eigenvalues. What is the trace of […]
• Powers of a Matrix Cannot be a Basis of the Vector Space of Matrices Let $n>1$ be a positive integer. Let $V=M_{n\times n}(\C)$ be the vector space over the complex numbers $\C$ consisting of all complex $n\times n$ matrices. The dimension of $V$ is $n^2$. Let $A \in V$ and consider the set \[S_A=\{I=A^0, A, A^2, \dots, A^{n^2-1}\}\] of $n^2$ […]
• Matrices Satisfying the Relation $HE-EH=2E$ Let $H$ and $E$ be $n \times n$ matrices satisfying the relation \[HE-EH=2E.\] Let $\lambda$ be an eigenvalue of the matrix $H$ such that the real part of $\lambda$ is the largest among the eigenvalues of $H$. Let $\mathbf{x}$ be an eigenvector corresponding to $\lambda$. Then […]
• Every Prime Ideal is Maximal if $a^n=a$ for any Element $a$ in the Commutative Ring Let $R$ be a commutative ring with identity $1\neq 0$. Suppose that for each element $a\in R$, there exists an integer $n > 1$ depending on $a$. Then prove that every prime ideal is a maximal ideal.   Hint. Let $R$ be a commutative ring with $1$ and $I$ be an ideal […]