# Boston-college-exam-eye-catch

• Sylow’s Theorem (Summary) In this post we review Sylow's theorem and as an example we solve the following problem. Show that a group of order $200$ has a normal Sylow $5$-subgroup. Review of Sylow's Theorem One of the important theorems in group theory is Sylow's theorem. Sylow's theorem is a […]
• Find an Orthonormal Basis of the Range of a Linear Transformation Let $T:\R^2 \to \R^3$ be a linear transformation given by $T\left(\, \begin{bmatrix} x_1 \\ x_2 \end{bmatrix} \,\right) = \begin{bmatrix} x_1-x_2 \\ x_2 \\ x_1+ x_2 \end{bmatrix}.$ Find an orthonormal basis of the range of $T$. (The Ohio […]
• The Quotient Ring by an Ideal of a Ring of Some Matrices is Isomorphic to $\Q$. Let $R=\left\{\, \begin{bmatrix} a & b\\ 0& a \end{bmatrix} \quad \middle | \quad a, b\in \Q \,\right\}.$ Then the usual matrix addition and multiplication make $R$ an ring. Let \[J=\left\{\, \begin{bmatrix} 0 & b\\ 0& 0 \end{bmatrix} […]
• Powers of a Diagonal Matrix Let $A=\begin{bmatrix} a & 0\\ 0& b \end{bmatrix}$. Show that (1) $A^n=\begin{bmatrix} a^n & 0\\ 0& b^n \end{bmatrix}$ for any $n \in \N$. (2) Let $B=S^{-1}AS$, where $S$ be an invertible $2 \times 2$ matrix. Show that $B^n=S^{-1}A^n S$ for any $n \in […] • A Group of Order$pqr$Contains a Normal Subgroup of Order Either$p, q$, or$r$Let$G$be a group of order$|G|=pqr$, where$p,q,r$are prime numbers such that$p<q<r$. Show that$G$has a normal subgroup of order either$p,q$or$r$. Hint. Show that using Sylow's theorem that$G$has a normal Sylow subgroup of order either$p,q$, or$r$. Review […] • The Preimage of Prime ideals are Prime Ideals Let$f: R\to R'$be a ring homomorphism. Let$P$be a prime ideal of the ring$R'$. Prove that the preimage$f^{-1}(P)$is a prime ideal of$R\$.   Proof. The preimage of an ideal by a ring homomorphism is an ideal. (See the post "The inverse image of an ideal by […]