Prove that two given groups are nonisomorphic
Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 2.3 Exercise 2.3.13 Prove that the following pairs of groups are not isomorphic: (1) $\mathbb{Z} \times Z_2$ and $\mathbb{Z}$,…
Prove that a given group is not cyclic
Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 2.3 Exercise 2.3.12 Prove that the following groups are not cyclic: (1) $Z_2 \times Z_2$, (2) $Z_2 \times \mathbb{Z}$,…
Prove that given subsets of a direct product are subgroups
Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 2.1 Exercise 2.1.11 Let $A$ and $B$ be groups. Prove that the following sets are subgroups of $A \times…
The coordinate projections from a direct product of groups are group homomorphisms
Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 1.6 Exercise 1.6.16 Let $A$ and $B$ be groups. Prove that the maps $\pi_1 : A \times B \rightarrow…
Compute the kernel of a coordinate projection
Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 1.6 Exercise 1.6.15 Define a map $\pi : \mathbb{R}^2 \rightarrow \mathbb{R}$ by $\pi((x,y)) = x$. Prove that $\pi$ is…
Direct product of groups is essentially associative
Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 1.6 Exercise 1.6.12 Let $A$, $B$, and $C$ be groups. Show that $A \times (B \times C) \cong (A…
Direct product of groups is essentially commutative
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:…
Compute the order of an element in a direct product of groups
Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 1.1 Exercise 1.1.30 Let $A$ and $B$ be groups and let $a \in A$ and $b \in B$ have…
A finite direct product is abelian if and only if each direct factor is abelian
Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 1.1 Exercise 1.1.29 Let $A$ and $B$ be groups. Prove that $A \times B$ is abelian if and only…
A finite direct product of groups is a group under componentwise multiplication
Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 1.1 Exercise 1.1.28 Let $A$ and $B$ be groups. Verify that $A \times B$ is a group under componentwise…