More from my site

• Basic Exercise Problems in Module Theory Let $R$ be a ring with $1$ and $M$ be a left $R$-module. (a) Prove that $0_Rm=0_M$ for all $m \in M$. Here $0_R$ is the zero element in the ring $R$ and $0_M$ is the zero element in the module $M$, that is, the identity element of the additive group $M$. To simplify the […]
• Welcome to Problems in Mathematics Welcome to my website. I post problems and their solutions/proofs in mathematics. Most of the problems are undergraduate level mathematics. Here are several topics I cover on this website. Topics Linear Algebra Group Theory Ring Theory Field Theory, Galois Theory Module […]
• Every Finite Group Having More than Two Elements Has a Nontrivial Automorphism Prove that every finite group having more than two elements has a nontrivial automorphism. (Michigan State University, Abstract Algebra Qualifying Exam)   Proof. Let $G$ be a finite group and $|G|> 2$. Case When $G$ is a Non-Abelian Group Let us first […]
• A Subgroup of Index a Prime $p$ of a Group of Order $p^n$ is Normal Let $G$ be a finite group of order $p^n$, where $p$ is a prime number and $n$ is a positive integer. Suppose that $H$ is a subgroup of $G$ with index $[G:P]=p$. Then prove that $H$ is a normal subgroup of $G$. (Michigan State University, Abstract Algebra Qualifying […]
• If Squares of Elements in a Group Lie in a Subgroup, then It is a Normal Subgroup Let $H$ be a subgroup of a group $G$. Suppose that for each element $x\in G$, we have $x^2\in H$. Then prove that $H$ is a normal subgroup of $G$. (Purdue University, Abstract Algebra Qualifying Exam)   Proof. To show that $H$ is a normal subgroup of […]
• Ring Homomorphisms from the Ring of Rational Numbers are Determined by the Values at Integers Let $R$ be a ring with unity. Suppose that $f$ and $g$ are ring homomorphisms from $\Q$ to $R$ such that $f(n)=g(n)$ for any integer $n$. Then prove that $f=g$.   Proof. Let $a/b \in \Q$ be an arbitrary rational number with integers $a, b$. Then we […]
• Determine Eigenvalues, Eigenvectors, Diagonalizable From a Partial Information of a Matrix Suppose the following information is known about a $3\times 3$ matrix $A$. \[A\begin{bmatrix} 1 \\ 2 \\ 1 \end{bmatrix}=6\begin{bmatrix} 1 \\ 2 \\ 1 \end{bmatrix}, \quad A\begin{bmatrix} 1 \\ -1 \\ 1 […]
• Conditional Probability When the Sum of Two Geometric Random Variables Are Known Let $X$ and $Y$ be geometric random variables with parameter $p$, with $0 \leq p \leq 1$. Assume that $X$ and $Y$ are independent. Let $n$ be an integer greater than $1$. Let $k$ be a natural number with $k\leq n$. Then prove the formula \[P(X=k \mid X + Y = n) = […]