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## The additive group of rational numbers has no subgroups of finite index

Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 3.2 Exercise 3.2.21 Prove that $\mathbb{Q}$ has no proper subgroups of finite index. Deduce that $\mathbb{Q}/\mathbb{Z}$ has no proper…

## Normal subgroups whose order and index are coprime are unique up to order

Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 3.2 Exercise 3.2.19 Let $G$ be a finite group, $N \leq G$ a normal subgroup, and suppose that $|N|$…

## 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…