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…
Laws of exponents in a group
Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 1.1 Exercise 1.1.19 Let $G$ be a group, $x \in G$, and $a,b \in \mathbb{Z}^+$. (1) Prove that $x^{a+b}…
Basic property of inverses of group elements of finite order
Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 1.1 Exercise 1.1.17 Let $G$ be a group and let $x \in G$. Prove that if $|x| = n$…
The inverse of a product is the reversed product of inverses
Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 1.1 Exercise 1.1.15 Let $G$ be a group. Prove that $$(a_1 \cdot \ldots \cdot a_n)^{-1} = a_n^{-1} \cdot \ldots…