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:…
If A and B have the same cardinality, then Sym(A) and Sym(B) are isomorphic
Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 1.6 Exercise 1.6.10 Let $\theta : \Delta \rightarrow \Omega$ be a bijection. Define $\varphi : S_\delta \rightarrow S_\Omega$ by…
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…