# Event_f_definition

• Centralizer, Normalizer, and Center of the Dihedral Group $D_{8}$ Let $D_8$ be the dihedral group of order $8$. Using the generators and relations, we have $D_{8}=\langle r,s \mid r^4=s^2=1, sr=r^{-1}s\rangle.$ (a) Let $A$ be the subgroup of $D_8$ generated by $r$, that is, $A=\{1,r,r^2,r^3\}$. Prove that the centralizer […]
• Subgroup of Finite Index Contains a Normal Subgroup of Finite Index Let $G$ be a group and let $H$ be a subgroup of finite index. Then show that there exists a normal subgroup $N$ of $G$ such that $N$ is of finite index in $G$ and $N\subset H$.   Proof. The group $G$ acts on the set of left cosets $G/H$ by left multiplication. Hence […]
• Find Eigenvalues, Eigenvectors, and Diagonalize the 2 by 2 Matrix Consider the matrix $A=\begin{bmatrix} a & -b\\ b& a \end{bmatrix}$, where $a$ and $b$ are real numbers and $b\neq 0$. (a) Find all eigenvalues of $A$. (b) For each eigenvalue of $A$, determine the eigenspace $E_{\lambda}$. (c) Diagonalize the matrix $A$ by finding a […]
• The Inverse Image of an Ideal by a Ring Homomorphism is an Ideal Let $f:R\to R'$ be a ring homomorphism. Let $I'$ be an ideal of $R'$ and let $I=f^{-1}(I)$ be the preimage of $I$ by $f$. Prove that $I$ is an ideal of the ring $R$.   Proof. To prove $I=f^{-1}(I')$ is an ideal of $R$, we need to check the following two […]
• The Number of Elements Satisfying $g^5=e$ in a Finite Group is Odd Let $G$ be a finite group. Let $S$ be the set of elements $g$ such that $g^5=e$, where $e$ is the identity element in the group $G$. Prove that the number of elements in $S$ is odd.   Proof. Let $g\neq e$ be an element in the group $G$ such that $g^5=e$. As […]
• Expected Value and Variance of Exponential Random Variable Let $X$ be an exponential random variable with parameter $\lambda$. (a) For any positive integer $n$, prove that $E[X^n] = \frac{n}{\lambda} E[X^{n-1}].$ (b) Find the expected value of $X$. (c) Find the variance of $X$. (d) Find the standard deviation of […]
• Conjugate of the Centralizer of a Set is the Centralizer of the Conjugate of the Set Let $X$ be a subset of a group $G$. Let $C_G(X)$ be the centralizer subgroup of $X$ in $G$. For any $g \in G$, show that $gC_G(X)g^{-1}=C_G(gXg^{-1})$.   Proof. $(\subset)$ We first show that $gC_G(X)g^{-1} \subset C_G(gXg^{-1})$. Take any $h\in C_G(X)$. Then for […]
• 7 Problems on Skew-Symmetric Matrices Let $A$ and $B$ be $n\times n$ skew-symmetric matrices. Namely $A^{\trans}=-A$ and $B^{\trans}=-B$. (a) Prove that $A+B$ is skew-symmetric. (b) Prove that $cA$ is skew-symmetric for any scalar $c$. (c) Let $P$ be an $m\times n$ matrix. Prove that $P^{\trans}AP$ is […]