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 $|x| = n$…
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 $$(a_1 \cdot \ldots \cdot a_n)^{-1} = a_n^{-1} \cdot \ldots…
Z/(n) is not a group under multiplication
Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 1.1 Exercise 1.1.5 Prove for all $n > 1$ that $\mathbb{Z}/(n)$ is not a group under multiplication of residue…
Solution to Mathematics for Machine Learning Exercise 2.1
We consider $(\mathbb R\setminus \{-1\},\star)$, where \begin{equation}\label{2.1.1}a\star b:= ab+a+b,\quad a,b\in\mathbb R\setminus \{-1\}\end{equation} a. Show that $(\mathbb R\setminus \{-1\},\star)$ is an Abelian group. b. Solve $$3\star x \star x = 15$$…
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