If you find any mistakes, please make a comment! Thank you.

In a division ring, every centralizer is a division ring

Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 7.1 Exercise 7.1.10 Prove that if $D$ is a division ring, then $C_D(a)$ is a division ring for all…

Compute the centralizers of each element in Sym(3), Dih(8), and the quaternion group

Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 2.2 Exercise 2.2.4 For each of the groups $S_3$, $D_8$, and $Q_8$, compute the centralizer of each element and…

Centralizer is inclusion-reversing

Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 2.2 Exercise 2.2.3 Let $G$ be a group. Prove that if $A$ and $B$ are subsets of $G$ with…

The centralizer and normalizer of a group center is the group itself

Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 2.2 Exercise 2.2.2 Let $G$ be a group. Prove that $C_G(Z(G)) = G$ and deduce that $N_G(Z(G)) = G$.…

An alternate characterization of centralizer

Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 2.2 Exercise 2.2.1 Prove that \$C_G(A) = \{ g \in G \ |\ g^{-1}ag = a\ \mathrm{for\ all}\ a…