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## Basic properties of Hall subgroups

Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 3.3 Exercise 3.3.10 Solution: We have $[HN:H \cap N] = [HN: H \cap N]$, so that [HN:N][N:H \cap N]…

## Interaction of Sylow subgroups with quotients

Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 3.3 Exercise 3.3.9 Let $p$ be a prime and let $G$ be a finite group of order $p^am$, where…

## The group of complex p-power roots of unity is a proper quotient of itself

Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 3.3 Exercise 3.3.8 Let $p$ be a prime and let $G$ be the group of $p-$power roots of 1 in…