# perfect-numbers

• The Ideal Generated by a Non-Unit Irreducible Element in a PID is Maximal Let $R$ be a principal ideal domain (PID). Let $a\in R$ be a non-unit irreducible element. Then show that the ideal $(a)$ generated by the element $a$ is a maximal ideal.   Proof. Suppose that we have an ideal $I$ of $R$ such that $(a) \subset I \subset […] • Group of Order pq is Either Abelian or the Center is Trivial Let G be a group of order |G|=pq, where p and q are (not necessarily distinct) prime numbers. Then show that G is either abelian group or the center Z(G)=1. Hint. Use the result of the problem "If the Quotient by the Center is Cyclic, then the Group is […] • Vector Space of Polynomials and a Basis of Its Subspace Let P_2 be the vector space of all polynomials of degree two or less. Consider the subset in P_2 \[Q=\{ p_1(x), p_2(x), p_3(x), p_4(x)\},$ where \begin{align*} &p_1(x)=1, &p_2(x)=x^2+x+1, \\ &p_3(x)=2x^2, &p_4(x)=x^2-x+1. \end{align*} (a) Use the basis $B=\{1, x, […] • Quiz 1. Gauss-Jordan Elimination / Homogeneous System. Math 2568 Spring 2017. (a) Solve the following system by transforming the augmented matrix to reduced echelon form (Gauss-Jordan elimination). Indicate the elementary row operations you performed. […] • Elements of Finite Order of an Abelian Group form a Subgroup Let$G$be an abelian group and let$H$be the subset of$G$consisting of all elements of$G$of finite order. That is, $H=\{ a\in G \mid \text{the order of a is finite}\}.$ Prove that$H$is a subgroup of$G$. Proof. Note that the identity element$e$of […] • The Center of a p-Group is Not Trivial Let$G$be a group of order$|G|=p^n$for some$n \in \N$. (Such a group is called a$p$-group.) Show that the center$Z(G)$of the group$G$is not trivial. Hint. Use the class equation. Proof. If$G=Z(G)$, then the statement is true. So suppose that$G\neq […]
• Find Values of $a$ so that Augmented Matrix Represents a Consistent System Suppose that the following matrix $A$ is the augmented matrix for a system of linear equations. $A= \left[\begin{array}{rrr|r} 1 & 2 & 3 & 4 \\ 2 &-1 & -2 & a^2 \\ -1 & -7 & -11 & a \end{array} \right],$ where $a$ is a real number. Determine all the […]
• The Range and Null Space of the Zero Transformation of Vector Spaces Let $U$ and $V$ be vector spaces over a scalar field $\F$. Define the map $T:U\to V$ by $T(\mathbf{u})=\mathbf{0}_V$ for each vector $\mathbf{u}\in U$. (a) Prove that $T:U\to V$ is a linear transformation. (Hence, $T$ is called the zero transformation.) (b) Determine […]