Every subgroup is contained in its normalizer
Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 2.2 Exercise 2.2.6 Let $G$ be a group and $H \leq G$. (1) Show that $H \leq N_G(H)$. Give…
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…
Set of all strictly upper triangular matrices is a subgroup of general linear group
Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 2.1 Exercise 2.1.17 Let $n \in \mathbb{Z}^+$ and let $F$ be a field.Prove that the set $$UT_n^1(F) = \{…
Set of invertible upper triangular matrices is a subgroup of general linear group
Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 2.1 Exercise 2.1.16 Let $n \in \mathbb{Z}^+$ and let $F$ be a field. Prove that the set $$UT_n(F) =…
The union of a chain of subgroups is a subgroup
Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 2.1 Exercise 2.1.15 Let $G$ be a group, and $\{ H_i \}_{i \in \mathbb{Z}}$ be an ascending chain of…
Set of all elements of a given order is not necessarily a subgroup
Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 2.1 Exercise 2.1.14 Let $n \geq 3$. Show that $\{ x \in D_{2n} \ |\ x^2 = 1\}$ is…