Consequences of equality among left and right cosets
Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 3.2 Exercise 3.2.6 Let $G$ be a group, $H \leq G$ a subgroup, and $g \in G$ a fixed…
Characterize the normalizer of a cyclic subgroup
Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 2.3 Exercise 2.3.24 Let $G$ be a finite group and let $x \in G$. (1) Prove that if $g…
Every normalizer contains the group center
Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 2.2 Exercise 2.2.11 Let $G$ be a group. Prove that $Z(G) \leq N_G(A)$ for any subset $A \subseteq G$.…
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…
Basic properties of normalizers with respect to a subgroup
Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 2.2 Exercise 2.2.9 Let $G$ be a group, $H \leq G$, and $A \subseteq G$. Define $$N_H(A) = \{…
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) =…
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 $N_G(Z(G)) = G$.…