Finite subgroups with relatively prime orders intersect trivially
Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 3.2 Exercise 3.2.8 Let $G$ be a group and let $H, K \leq G$ be finite subgroups of relatively…
Compute the subgroup lattice of Z/(48)
Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 2.3 Exercise 2.3.6 In $\mathbb{Z}/(48)$, write out all elements of $\langle \overline{a} \rangle$ for every $\overline{a}$. Find all inclusions…
Compute the subgroup lattice of Z/(45)
Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 2.3 Exercise 2.3.1 Find all subgroups of $G = \mathbb{Z}/(45)$, giving a generator for each. Describe the containments among…
2×2 invertible upper triangular matrices form a subgroup of general linear group
Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 1.4 Exercise 1.4.10 Let $G = \left\{ \left[{a \atop 0} {b \atop c}\right] \ |\ a,b,c \in \mathbb{R}, a…
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 $$N_H(A) = \{…
Every subgroup is contained in its normalizer
Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 2.2 Exercise 2.2.6 Let $G$ be a group and $H \leq G$. (1) Show that $H \leq N_G(H)$. Give…
Set of all strictly upper triangular matrices is a subgroup of general linear group
Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 2.1 Exercise 2.1.17 Let $n \in \mathbb{Z}^+$ and let $F$ be a field.Prove that the set $$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) =…