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• Generators of the Augmentation Ideal in a Group Ring Let $R$ be a commutative ring with $1$ and let $G$ be a finite group with identity element $e$. Let $RG$ be the group ring. Then the map $\epsilon: RG \to R$ defined by \[\epsilon(\sum_{i=1}^na_i g_i)=\sum_{i=1}^na_i,\] where $a_i\in R$ and $G=\{g_i\}_{i=1}^n$, is a ring […]
• Group Homomorphism, Conjugate, Center, and Abelian group Let $G$ be a group. We fix an element $x$ of $G$ and define a map \[ \Psi_x: G\to G\] by mapping $g\in G$ to $xgx^{-1} \in G$. Then prove the followings. (a) The map $\Psi_x$ is a group homomorphism. (b) The map $\Psi_x=\id$ if and only if $x\in Z(G)$, where $Z(G)$ is the […]
• Jewelry Company Quality Test Failure Probability A jewelry company requires for its products to pass three tests before they are sold at stores. For gold rings, 90 % passes the first test, 85 % passes the second test, and 80 % passes the third test. If a product fails any test, the product is thrown away and it will not take the […]
• Non-Example of a Subspace in 3-dimensional Vector Space $\R^3$ Let $S$ be the following subset of the 3-dimensional vector space $\R^3$. \[S=\left\{ \mathbf{x}\in \R^3 \quad \middle| \quad \mathbf{x}=\begin{bmatrix} x_1 \\ x_2 \\ x_3 \end{bmatrix}, x_1, x_2, x_3 \in \Z \right\}, \] where $\Z$ is the set of all integers. […]
• The Subspace of Linear Combinations whose Sums of Coefficients are zero Let $V$ be a vector space over a scalar field $K$. Let $\mathbf{v}_1, \mathbf{v}_2, \dots, \mathbf{v}_k$ be vectors in $V$ and consider the subset \[W=\{a_1\mathbf{v}_1+a_2\mathbf{v}_2+\cdots+ a_k\mathbf{v}_k \mid a_1, a_2, \dots, a_k \in K \text{ and } […]
• Solving a System of Differential Equation by Finding Eigenvalues and Eigenvectors Consider the system of differential equations \begin{align*} \frac{\mathrm{d} x_1(t)}{\mathrm{d}t} & = 2 x_1(t) -x_2(t) -x_3(t)\\ \frac{\mathrm{d}x_2(t)}{\mathrm{d}t} & = -x_1(t)+2x_2(t) -x_3(t)\\ \frac{\mathrm{d}x_3(t)}{\mathrm{d}t} & = -x_1(t) -x_2(t) […]
• 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 […]
• The Quadratic Integer Ring $\Z[\sqrt{-5}]$ is not a Unique Factorization Domain (UFD) Prove that the quadratic integer ring $\Z[\sqrt{-5}]$ is not a Unique Factorization Domain (UFD). Proof. Any element of the ring $\Z[\sqrt{-5}]$ is of the form $a+b\sqrt{-5}$ for some integers $a, b$. The associated (field) norm $N$ is given […]