If you find any mistakes, please make a comment! Thank you.

## Right multiplication by the inverse is a left group action of a group on itself

Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 1.7 Exercise 1.7.15 Let $G$ be any group. Show that the mapping defined by $g \cdot a = ag^{-1}$…

## Counterexamples regarding one-sided zero divisors and inverses

Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 7.1 Exercise 7.1.30 Let $A = \prod_\mathbb{N} \mathbb{Z}$ be the direct product of countably many copies of $\mathbb{Z}$. Recall…

## Basic properties of left and right units and left and right zero divisors

Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 7.1 Exercise 7.1.28 Let $R$ be a ring with $1 \neq 0$. A nonzero element $a \in R$ is…

## 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…

## The inversion map is a homomorphism precisely on abelian groups

Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 1.6 Exercise 1.6.17 Let $G$ be a group. Prove that the map $\varphi : G \rightarrow G$ given by…

## Every group element of odd order is an odd power of its square

Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 1.1 Exercise 1.1.21 Let $G$ be a finite group and let $x \in G$ be an element of order…

## A group element and its inverse have the same order

Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 1.1 Exercise 1.1.20 Let $G$ be a group and let $x \in G$. Prove that $x$ and $x^{-1}$ have…