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…
The kernel of a group homomorphism is a normal subgroup
Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 3.1 Exercise 3.1.1 Let $\varphi : G \rightarrow H$ be a group homomorphism and let $E \leq H$ be…
Characterize the automorphisms of cyclic group of order n
Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 2.3 Exercise 2.3.26 Let $Z_n = \langle \alpha \rangle$ be a cyclic group of order $n$ and for each…
On finite groups, power maps with exponent relatively prime to the group order are surjective
Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 2.3 Exercise 2.3.25 Let $G$ be a cyclic group of order $n$ and let $k$ be an integer relatively…
Characterize the normalizer of a cyclic subgroup
Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 2.3 Exercise 2.3.24 Let $G$ be a finite group and let $x \in G$. (1) Prove that if $g…
The group of units in $\mathbb Z/(2^n)$ is not cyclic for n at least 3
Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 2.3 Exercise 2.3.23 Show that $(\mathbb{Z}/(2^n))^\times$ is not cyclic for any $n \geq 3$. (Hint: find two distinct subgroups…
Compute the order of 5 in the integers mod a power of 2
Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 2.3 Exercise 2.3.22 Let $n$ be an integer with $n \geq 3$. Use the Binomial Theorem to show that…