Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 5.1 Exercise 5.1.9 Solution: We know that if $F$ is a finite field then $\mathsf{Aut}(F^n) \cong GL_n(F)$. This isomorphism…
Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 3.1 Exercise 3.1.35 Let $F$ be a field and $n$ a positive integer. Prove that $SL_n(F)$ is normal in…
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.…
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…
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…
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 -…
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 $|GL_n(F)| < q^{n^2}$.…
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…
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…
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…