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) $Z_2 \times \mathbb{Z}$,…
Exhibit the cyclic subgroups of Dih(8) as sets
Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 2.3 Exercise 2.3.11 Find all cyclic subgroups of $D_8$. Exhibit a proper subgroup of $D_8$ which is not cyclic.…
If the group and an element have the same finite order then the group is cyclic
Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 2.3 Exercise 2.3.2 Let $G$ be a finite group and let $x \in G$. Prove that if $|x| =…
If n is composite, then Z/(n) is not a field
Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 1.4 Exercise 1.4.4 Show that if $n$ is not prime, then $\mathbb{Z}/(n)$ is not a field. Solution: If $n$…
Show that a given general linear group is nonabelian
Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 1.4 Exercise 1.4.3 Show that $GL_2(\mathbb{F}_2)$ is non-abelian. Solution: We have $$\left[ {1 \atop 1}{1 \atop 0} \right] \cdot…
Every subgroup is contained in its normalizer
Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 2.2 Exercise 2.2.6 Let $G$ be a group and $H \leq G$. (1) Show that $H \leq N_G(H)$. Give…
Compute a torsion subgroup
Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 2.1 Exercise 2.1.7 Fix $n \in \mathbb{Z}^+$ with $n > 1$. Find the torsion subgroup of $\mathbb{Z} \times \mathbb{Z}/(n)$.…
Torsion elements in an abelian group form a subgroup
Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 2.1 Exercise 2.1.6 Let $G$ be a group. An element $x \in G$ is called torsion if it has…
Exhibit an infinite subset in a group which is multiplicatively closed but not a subgroup
Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 2.1 Exercise 2.1.4 Give an explicit example of a group $G$ and an infinite subset $H$ of $G$ which…