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…
Basic properties of centralizers in a ring
Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 7.1 Exercise 7.1.9 Let $R$ be a ring. For a fixed element $a \in R$, define $C_R(a) = \{r…
Normal subgroups of order 2 are central
Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 2.2 Exercise 2.2.10 Let $H$ be a subgroup of order 2 in $G$. Show that $N_G(H) = C_G(H)$. Deduce…
Demonstrate that a given subgroup is normal
Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 2.2 Exercise 2.2.5 For each of the following subgroups $A$ of a given group $G$, show that $C_G(A) =…
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…