# group theory in mathematics

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- Welcome 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 […]
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- 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 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 […]
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- Compute 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 […]