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## 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…

## 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 $n$…

## Compute the order of each element in the quaternion group

Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 1.5 Exercise 1.5.1 Compute the order of each of the elements in $Q_8$. Solution: $x$            Reasoning            Order 1 1…