# Tagged: Abelian Group

## Q/Z is divisible

Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 3.1 Exercise 3.1.15 Prove that a quotient of a divisible abelian group by any proper subgroup is also divisible. Deduce that $\mathbb{Q}/\mathbb{Z}$ is...

## Every quotient of an abelian group is abelian

Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 3.1 Exercise 3.1.3 Let $A$ be an abelian group and let $B \leq A$. Prove that $A/B$ is abelian. Give an example of...

## Additive subgroups of the rationals which are closed under inversion are trivial

Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 2.1 Exercise 2.1.13 Let $H$ be a subgroup of the additive group of rational numbers with the property that if $x \in H$...

## The n-th powers and n-th roots of an abelian group are subgroups

Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 2.1 Exercise 2.1.12 Let $A$ be an abelian group and fix $n \in \mathbb{Z}^+$. Prove that the following subsets are subgroups of $A$....

## Compute a torsion subgroup

Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 2.1 Exercise 2.1.7 Fix $n \in \mathbb{Z}^+$ with $n > 1$. Find the torsion subgroup of $\mathbb{Z} \times \mathbb{Z}/(n)$. Show that the set...

## Torsion elements in an abelian group form a subgroup

Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 2.1 Exercise 2.1.6 Let $G$ be a group. An element $x \in G$ is called torsion if it has finite order. The set...

## If a group has an automorphism which is fixed point free of order 2, then it is abelian

Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 1.6 Exercise 1.6.23 Let $G$ be a finite group which possesses an automorphism $\sigma$ such that $\sigma(g) = g$ if and only if...

## The square map is a group homomorphism precisely on abelian groups

Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 1.6 Exercise 1.6.18 Let $G$ be a group. Show that the map $\varphi : G \rightarrow G$ given by $g \mapsto g^2$ is...