Compute the index of the special linear group in the general linear group
Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 3.3 Exercise 3.3.1 Let $F$ be a finite field of order $q$ and let $n$ be a positive integer.…
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…
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…
Compute the order of each element in the general linear group of dimension 2 over Z/(2)
Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 1.4 Exercise 1.4.2 Write out all the elements in $GL_2(\mathbb{F}_2)$ and compute the order of each element. Solution: We…
Compute the number of invertible 2×2 matrices over Z/(2)
Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 1.4 Exercise 1.4.1 Prove that $|GL_2(\mathbb{F}_2)| = 6$. Solution: $GL_2(\mathbb{F}_2$ consists of precisely those $2 \times 2$ matrices over…