## Conjugation by a fixed group element is an automorphism

Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 1.7 Exercise 1.7.17 Let $G$ be a group and let $G$ act on itself by left conjugation; i.e., $g…

Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 1.7 Exercise 1.7.17 Let $G$ be a group and let $G$ act on itself by left conjugation; i.e., $g…

Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 7.1 Exercise 7.1.29 Let $A$ be any abelian group. Let $R = \mathsf{Hom}(A,A)$ be the set of all group…

Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 2.3 Exercise 2.3.26 Let $Z_n = \langle \alpha \rangle$ be a cyclic group of order $n$ and for each…

Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 1.6 Exercise 1.6.23 Let $G$ be a finite group which possesses an automorphism $\sigma$ such that $\sigma(g) = g$…

Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 1.6 Exercise 1.6.22 Let $A$ be an abelian group and fix some $k \in \mathbb{Z}$. Prove that the map…

Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 1.6 Exercise 1.6.21 Prove that for each fixed nonzero $k \in \mathbb{Q}$ the map $\varphi : \mathbb{Q} \rightarrow \mathbb{Q}$…

Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 1.6 Exercise 1.6.20 Let $G$ be a group and let $\mathsf{Aut}(G)$ be the set of all isomorphisms $G \rightarrow…