normal-subgroup

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Normal Subgroups Problems and Solutions in Group Theory


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  • Three Equivalent Conditions for a Ring to be a FieldThree Equivalent Conditions for a Ring to be a Field Let $R$ be a ring with $1$. Prove that the following three statements are equivalent. The ring $R$ is a field. The only ideals of $R$ are $(0)$ and $R$. Let $S$ be any ring with $1$. Then any ring homomorphism $f:R \to S$ is injective.   Proof. […]
  • Give the Formula for a Linear Transformation from $\R^3$ to $\R^2$Give the Formula for a Linear Transformation from $\R^3$ to $\R^2$ Let $T: \R^3 \to \R^2$ be a linear transformation such that \[T(\mathbf{e}_1)=\begin{bmatrix} 1 \\ 4 \end{bmatrix}, T(\mathbf{e}_2)=\begin{bmatrix} 2 \\ 5 \end{bmatrix}, T(\mathbf{e}_3)=\begin{bmatrix} 3 \\ 6 […]
  • Solve a Linear Recurrence Relation Using Vector Space TechniqueSolve a Linear Recurrence Relation Using Vector Space Technique Let $V$ be a real vector space of all real sequences \[(a_i)_{i=1}^{\infty}=(a_1, a_2, \dots).\] Let $U$ be a subspace of $V$ defined by \[U=\{(a_i)_{i=1}^{\infty}\in V \mid a_{n+2}=2a_{n+1}+3a_{n} \text{ for } n=1, 2,\dots \}.\] Let $T$ be the linear transformation from […]
  • A Group of Order the Square of a Prime is AbelianA Group of Order the Square of a Prime is Abelian Suppose the order of a group $G$ is $p^2$, where $p$ is a prime number. Show that (a) the group $G$ is an abelian group, and (b) the group $G$ is isomorphic to either $\Zmod{p^2}$ or $\Zmod{p} \times \Zmod{p}$ without using the fundamental theorem of abelian […]
  • Submodule Consists of Elements Annihilated by Some Power of an IdealSubmodule Consists of Elements Annihilated by Some Power of an Ideal Let $R$ be a ring with $1$ and let $M$ be an $R$-module. Let $I$ be an ideal of $R$. Let $M'$ be the subset of elements $a$ of $M$ that are annihilated by some power $I^k$ of the ideal $I$, where the power $k$ may depend on $a$. Prove that $M'$ is a submodule of […]
  • Idempotent Matrices are DiagonalizableIdempotent Matrices are Diagonalizable Let $A$ be an $n\times n$ idempotent matrix, that is, $A^2=A$. Then prove that $A$ is diagonalizable.   We give three proofs of this problem. The first one proves that $\R^n$ is a direct sum of eigenspaces of $A$, hence $A$ is diagonalizable. The second proof proves […]
  • Inverse Matrix of Positive-Definite Symmetric Matrix is Positive-DefiniteInverse Matrix of Positive-Definite Symmetric Matrix is Positive-Definite Suppose $A$ is a positive definite symmetric $n\times n$ matrix. (a) Prove that $A$ is invertible. (b) Prove that $A^{-1}$ is symmetric. (c) Prove that $A^{-1}$ is positive-definite. (MIT, Linear Algebra Exam Problem)   Proof. (a) Prove that $A$ is […]
  • Example of a Nilpotent Matrix $A$ such that $A^2\neq O$ but $A^3=O$.Example of a Nilpotent Matrix $A$ such that $A^2\neq O$ but $A^3=O$. Find a nonzero $3\times 3$ matrix $A$ such that $A^2\neq O$ and $A^3=O$, where $O$ is the $3\times 3$ zero matrix. (Such a matrix is an example of a nilpotent matrix. See the comment after the solution.)   Solution. For example, let $A$ be the following $3\times […]

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