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…
Definition of the Heisenberg group over a field
Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 1.4 Exercise 1.4.11 Let $F$ be a field, and define the Heisenberg group $H(F)$ over, $F$ by (1) Show…
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…
Multiplication of 2×2 matrices over the real numbers is associative
Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 1.4 Exercise 1.4.9 Prove that the binary operation of matrix multiplication of $2 \times 2$ matrices over $\mathbb{R}$ is…
General linear groups of dimension at least 2 are nonabelian
Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 1.4 Exercise 1.4.8 Show that $GL_n(F)$ is nonabelian for all $n \geq 2$ and all fields $F$. Solution: Recall…
Computation of the order of the general linear group of index 2 over Z/(p)
Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 1.4 Exercise 1.4.7 Let $p$ be a prime. Prove that the order of $GL_2(\mathbb{F}_p)$ is $p^4 - p^3 -…
The order of a general linear group over a finite field is bounded above
Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 1.4 Exercise 1.4.6 Let $F$ be a field. If $|F| = q$ is finite show that $|GL_n(F)| < q^{n^2}$.…
A general linear group over a field is finite if and only if the field is finite
Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 1.4 Exercise 1.4.5 Let $F$ be a field. Show that $GL_n(F)$ is a finite group if and only if…
If n is composite, then Z/(n) is not a field
Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 1.4 Exercise 1.4.4 Show that if $n$ is not prime, then $\mathbb{Z}/(n)$ is not a field. Solution: If $n$…
Show that a given general linear group is nonabelian
Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 1.4 Exercise 1.4.3 Show that $GL_2(\mathbb{F}_2)$ is non-abelian. Solution: We have $$\left[ {1 \atop 1}{1 \atop 0} \right] \cdot…