Dih(24) and Sym(4) are not isomorphic
Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 1.6 Exercise 1.6.9 Prove that $D_{24}$ and $S_4$ are not isomorphic. Solution: We know from Exercise 1.2.3 that $D_{24}$…
Finite symmetric groups of different orders are not isomorphic
Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 1.6 Exercise 1.6.8 Prove that if $n \neq m$ then $S_n$ and $S_m$ are not isomorphic. Solution: We know…
Dih(8) and the quaternion group are not isomorphic
Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 1.6 Exercise 1.6.7 Prove that $D_8$ and $Q_8$ are not isomorphic. Solution: We saw in Exercise 1.5.2 that $D_8$…
The additive groups of integers and rational numbers are not isomorphic
Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 1.6 Exercise 1.6.6 Prove that the additive groups $\mathbb{Z}$ and $\mathbb{Q}$ are not isomorphic. Solution: First we prove a…
The additive groups of rational and real numbers are not isomorphic
Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 1.6 Exercise 1.6.5 Prove that the additive groups $\mathbb{Q}$ and $\mathbb{R}$ are not isomorphic. Solution: We know that no…
The multiplicative groups of nonzero real numbers and nonzero complex numbers are not isomorphic
Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 1.6 Exercise 1.6.4 Prove that the multiplicative groups $\mathbb{R}^\times$ and $\mathbb{C}^\times$ are not isomorphic. Solution: Recall from Exercise 1.6.2…
Surjective group homomorphisms preserve abelianicity
Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 1.6 Exercise 1.6.3 If $\varphi : G \rightarrow H$ is an isomorphism, prove that $G$ is abelian if and…
Group isomorphisms preserve the order of an element
Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 1.6 Exercise 1.6.2 Let $G$ and $H$ be groups. If $\varphi : G \rightarrow H$ is an isomorphism, show…
Group homomorphisms preserve exponents
Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 1.6 Exercise 1.6.1 Let $G$ and $H$ be groups and $\varphi : G \rightarrow H$ a group homomorphism. (1)…
Detecting copies of Dih(2n) in a larger group
Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 1.2 Exercise 1.2.6 Let $G$ be a group, and let $x,y \in G$ be elements of order 2. Prove…