Find a group presentation for cyclic group of order n
Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 2.3 Exercise 2.3.17 Find a presentation for $Z_n $ with one generator. Solution: We have $Z_n = \langle x…
Prove that two given groups are nonisomorphic
Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 2.3 Exercise 2.3.13 Prove that the following pairs of groups are not isomorphic: (1) $\mathbb{Z} \times Z_2$ and $\mathbb{Z}$,…
Prove that a given group is not cyclic
Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 2.3 Exercise 2.3.12 Prove that the following groups are not cyclic: (1) $Z_2 \times Z_2$, (2) $Z_2 \times \mathbb{Z}$,…
Exhibit the cyclic subgroups of Dih(8) as sets
Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 2.3 Exercise 2.3.11 Find all cyclic subgroups of $D_8$. Exhibit a proper subgroup of $D_8$ which is not cyclic.…
Compute the order of a cyclic subgroup in Z/(54)
Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 2.3 Exercise 2.3.10 What is the order of $\overline{30}$ in $\mathbb{Z}/(54)$? Write out all of the elements and their…
When is a map from Z/(48) to Cyc(36) a group homomorphism?
Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 2.3 Exercise 2.3.9 Let $Z_{36} = \langle x \rangle$. For which integers $a$ does the map $\psi_a$ defined by…
Exhibit the automorphisms of Z/(48)
Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 2.3 Exercise 2.3.8 Let $Z_{48} = \langle x \rangle$. For which integers a does the map $\varphi_a$ defined by…
Find the subgroup lattice of the cyclic group of order 48
Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 2.3 Exercise 2.3.7 Let $Z_{48} = \langle x \rangle$ and use the isomorphism $\mathbb{Z}/(48) \cong Z_{48}$ given by $\overline{1}…
Compute the subgroup lattice of Z/(48)
Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 2.3 Exercise 2.3.6 In $\mathbb{Z}/(48)$, write out all elements of $\langle \overline{a} \rangle$ for every $\overline{a}$. Find all inclusions…
Compute the number of generators of Z/(49000)
Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 2.3 Exercise 2.3.5 Find the number of generators for $\mathbb{Z}/(49000)$. Solution: The number of generators of $\mathbb{Z}/(n)$ is $\varphi(n)$,…