group theory in mathematics

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  • Welcome to Problems in MathematicsWelcome to Problems in Mathematics Welcome to my website. I post problems and its solutions/proofs in mathematics almost every day. 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 […]
  • Basic Exercise Problems in Module TheoryBasic 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 […]
  • Number Theoretical Problem Proved by Group Theory. $a^{2^n}+b^{2^n}\equiv 0 \pmod{p}$ Implies $2^{n+1}|p-1$.Number Theoretical Problem Proved by Group Theory. $a^{2^n}+b^{2^n}\equiv 0 \pmod{p}$ Implies $2^{n+1}|p-1$. Let $a, b$ be relatively prime integers and let $p$ be a prime number. Suppose that we have \[a^{2^n}+b^{2^n}\equiv 0 \pmod{p}\] for some positive integer $n$. Then prove that $2^{n+1}$ divides $p-1$.   Proof. Since $a$ and $b$ are relatively prime, at least one […]
  • Ring Homomorphisms from the Ring of Rational Numbers are Determined by the Values at IntegersRing 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 […]
  • Sylow’s Theorem (Summary)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 […]
  • The Center of a p-Group is Not TrivialThe Center of a p-Group is Not Trivial Let $G$ be a group of order $|G|=p^n$ for some $n \in \N$. (Such a group is called a $p$-group.) Show that the center $Z(G)$ of the group $G$ is not trivial. Hint. Use the class equation. Proof. If $G=Z(G)$, then the statement is true. So suppose that $G\neq […]
  • Mathematics About the Number 2017Mathematics About the Number 2017 Happy New Year 2017!! Here is the list of mathematical facts about the number 2017 that you can brag about to your friends or family as a math geek. 2017 is a prime number Of course, I start with the fact that the number 2017 is a prime number. The previous prime year was […]
  • Compute the Product  $A^{2017}\mathbf{u}$ of a Matrix Power and a VectorCompute the Product $A^{2017}\mathbf{u}$ of a Matrix Power and a Vector Let \[A=\begin{bmatrix} -1 & 2 \\ 0 & -1 \end{bmatrix} \text{ and } \mathbf{u}=\begin{bmatrix} 1\\ 0 \end{bmatrix}.\] Compute $A^{2017}\mathbf{u}$.   (The Ohio State University, Linear Algebra Exam) Solution. We first compute $A\mathbf{u}$. We […]

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