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 $|x| = n$…
Characterization of group elements whose square is the identity
Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 1.1 Exercise 1.1.16 Let $G$ be a group and let $x \in G$. Prove that $x^2 = 1$ if…
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…
Compute multiplicative orders in Z/(36)
Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 1.1 Exercise 1.1.14 Find the orders of the following elements of the multiplicative group $(\mathbb{Z}/(36))^\times$: $\overline{1}$, $\overline{-1}$, $\overline{5}$, $\overline{13}$,…
Compute additive orders in Z/(n)
Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 1.1 Exercise 1.1.13 Find the orders of the following elements of the additive group $\mathbb{Z}/(36)$: $\overline{1}$, $\overline{2}$, $\overline{6}$, $\overline{9}$,…
Compute multiplicative orders in Z/(n)
Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 1.1 Exercise 1.1.12 Find the orders of the following elements of the multiplicative group $(\mathbb{Z}/(12))^\times$: $\overline{1}$, $\overline{-1}$, $\overline{5}$, $\overline{7}$,…
The orders of the elements in a cyclic group of order 12
Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 1.1 Exercise 1.1.11 Find the orders of each element of the additive group $\mathbb{Z}/(12)$. Solution: For an element $n$…
A finite group is abelian if and only if its Cayley table is a symmetric matrix
Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 1.1 Exercise 1.1.10 Prove that a finite group is abelian if and only if its group table is a…
The rational numbers with the square root of 2 adjoined form a group
Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 1.1 Exercise 1.1.9 Let $G = \{ a+b \sqrt{2} \in \mathbb{R} \ |\ a,b \in \mathbb{Q} \}$. (1) Prove…