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## If the index and order of a normal subgroup and subgroup are relatively prime, then the subgroup is contained in the normal subgroup

Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 3.2 Exercise 3.2.18 Let $G$ be a group and let $H,N \leq G$ with $N$ normal in $G$. Prove…

## Finite subgroups with relatively prime orders intersect trivially

Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 3.2 Exercise 3.2.8 Let $G$ be a group and let $H, K \leq G$ be finite subgroups of relatively…

## A finite group of width two has a trivial center

Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 3.2 Exercise 3.2.4 Let $G$ be a group. Prove that if $|G| = pq$ for some primes $p$ and…

## Permissible subgroup orders in a group of order 120, and their indices

Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 3.2 Exercise 3.2.1 Which of the following are permissible orders for subgroups of a group of order 120: 1,…