# abelian-group-eye-catch

by Yu · Published · Updated

Add to solve later

Add to solve later

Add to solve later

### More from my site

- 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 […]
- Find Values of $h$ so that the Given Vectors are Linearly Independent Find the value(s) of $h$ for which the following set of vectors \[\left \{ \mathbf{v}_1=\begin{bmatrix} 1 \\ 0 \\ 0 \end{bmatrix}, \mathbf{v}_2=\begin{bmatrix} h \\ 1 \\ -h \end{bmatrix}, \mathbf{v}_3=\begin{bmatrix} 1 \\ 2h \\ 3h+1 […]
- Is the Derivative Linear Transformation Diagonalizable? Let $\mathrm{P}_2$ denote the vector space of polynomials of degree $2$ or less, and let $T : \mathrm{P}_2 \rightarrow \mathrm{P}_2$ be the derivative linear transformation, defined by \[ T( ax^2 + bx + c ) = 2ax + b . \] Is $T$ diagonalizable? If so, find a diagonal matrix which […]
- Determinant of Matrix whose Diagonal Entries are 6 and 2 Elsewhere Find the determinant of the following matrix \[A=\begin{bmatrix} 6 & 2 & 2 & 2 &2 \\ 2 & 6 & 2 & 2 & 2 \\ 2 & 2 & 6 & 2 & 2 \\ 2 & 2 & 2 & 6 & 2 \\ 2 & 2 & 2 & 2 & 6 \end{bmatrix}.\] (Harvard University, Linear Algebra Exam […]
- Linear Algebra Midterm 1 at the Ohio State University (3/3) The following problems are Midterm 1 problems of Linear Algebra (Math 2568) at the Ohio State University in Autumn 2017. There were 9 problems that covered Chapter 1 of our textbook (Johnson, Riess, Arnold). The time limit was 55 minutes. This post is Part 3 and contains […]
- Subset of Vectors Perpendicular to Two Vectors is a Subspace Let $\mathbf{a}$ and $\mathbf{b}$ be fixed vectors in $\R^3$, and let $W$ be the subset of $\R^3$ defined by \[W=\{\mathbf{x}\in \R^3 \mid \mathbf{a}^{\trans} \mathbf{x}=0 \text{ and } \mathbf{b}^{\trans} \mathbf{x}=0\}.\] Prove that the subset $W$ is a subspace of […]
- Injective Group Homomorphism that does not have Inverse Homomorphism Let $A=B=\Z$ be the additive group of integers. Define a map $\phi: A\to B$ by sending $n$ to $2n$ for any integer $n\in A$. (a) Prove that $\phi$ is a group homomorphism. (b) Prove that $\phi$ is injective. (c) Prove that there does not exist a group homomorphism $\psi:B […]
- Are Groups of Order 100, 200 Simple? Determine whether a group $G$ of the following order is simple or not. (a) $|G|=100$. (b) $|G|=200$. Hint. Use Sylow's theorem and determine the number of $5$-Sylow subgroup of the group $G$. Check out the post Sylow’s Theorem (summary) for a review of Sylow's […]