## Basic properties of n-ary direct products of groups

Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 5.1 Exercise 5.1.16 Solution: (1) First we show that $\iota_k$ is an injective homomorphism. If $g,h \in G_k$, then…

Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 5.1 Exercise 5.1.16 Solution: (1) First we show that $\iota_k$ is an injective homomorphism. If $g,h \in G_k$, then…

Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 5.1 Exercise 5.1.3 Solution: Let $x = (x_j)$ and $y = (y_j)$. If $j \in I$, then $$(xy)_j =…

Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 7.1 Exercise 7.1.21 Let $X$ be a nonempty set and let $R = \mathcal{P}(X)$ denote the power set of…

Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 7.1 Exercise 7.1.17 Let $R$ and $S$ be rings. Prove that the direct product $R \times S$ is a…

Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 7.1 Exercise 7.1.15 A ring $R$ is called Boolean if $a^2 = a$ for all $a \in R$. Prove…

Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 2.3 Exercise 2.3.16 Let $G$ be a group with $x,y \in G$. Assume $|x| = n$ and $|y| =…

Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 1.6 Exercise 1.6.11 Let $A$ and $B$ be groups. Prove that $A \times B \cong B \times A$. Solution:…

Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 1.1 Exercise 1.1.24 Let $G$ be a group and let $a,b \in G$ such that $ab = ba$. Prove…