# Category: Abstract Algebra Dummit Foote

## The special linear group is normal in the general linear group

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 $GL_n(F)$ and describe the...

## Perform computations in a quotient of dihedral group of order 16

Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 3.1 Exercise 3.1.17 Let $G = D_{16}$ and let $H = \langle r^4 \rangle$. (1) Show that the order of $G/H$ is 8....

## The quotient of a generating set is a generating set of the quotient

Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 3.1 Exercise 3.1.16 Let $G$ be a group and $N \leq G$ a normal subgroup. Show that if $G = \langle S \rangle$,...

## Q/Z is divisible

Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 3.1 Exercise 3.1.15 Prove that a quotient of a divisible abelian group by any proper subgroup is also divisible. Deduce that $\mathbb{Q}/\mathbb{Z}$ is...