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…
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.…
Q/Z is isomorphic to the group of complex roots of unity
Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 3.1 Exercise 3.1.14 Consider the additive quotient group $\mathbb{Q}/\mathbb{Z}$. (1) Show that every coset of $\mathbb{Z}$ in $\mathbb{Q}/\mathbb{Z}$ has…
Compute the order of a quotient group element
Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 3.1 Exercise 3.1.5 Let $G$ be a group and $N$ a normal subgroup of $G$. Prove that the order…
Powers in a quotient group
Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 3.1 Exercise 3.1.4 Let $G$ be a group and $N$ a normal subgroup of $G$. Show that for all…
Every quotient of an abelian group is abelian
Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 3.1 Exercise 3.1.3 Let $A$ be an abelian group and let $B \leq A$. Prove that $A/B$ is abelian.…
A fact about preimages under group homomorphisms
Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 3.1 Exercise 3.1.2 Let $\varphi : G \rightarrow H$ be a group homomorphism with kernel $K$ and let $a,b…
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