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## n divides the totient of $p^n-1$ when p is prime

Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 3.2 Exercise 3.2.15 Let $p$ be a prime and let $n$ be a positive integer. Find the order of…

## Alternate proof of Cauchy’s Theorem

Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 3.2 Exercise 3.2.9 Let $G$ be a finite group and let $p$ be a prime dividing $|G|$. Let $\mathcal{S}$…

## Examples of nilpotent elements

Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 7.1 Exercise 7.1.13 An element $x \in R$, $R$ a ring, is called nilpotent if $x^m = 0$ for some…