The kernel of a group homomorphism is a subgroup
Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 1.6 Exercise 1.6.14 Let $\varphi : G \rightarrow H$ be a group homomorphism. Define the kernel of $\varphi$ to…
The image of a group homomorphism is a subgroup
Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 1.6 Exercise 1.6.13 Let $G$ and $H$ be groups and let $\varphi : G \rightarrow H$ be a group…
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 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)…