Problem 1
All the quotient groups are cyclic and therefore isomorphic to \(\mathbb{Z}_{n}\) for some \(n\). In each case, find this \(n\). $$ \mathbb{Z}_{6} /\langle 2\rangle $$
Problem 1
Find all the cosets of the subgroup \(5 Z\) in \(\mathbb{Z}\).
Problem 1
Determine whether the indicated subgroup is normal in the indicated group. $$ A_{3} \text { in } S_{3} $$
Problem 1
In Exercises 1 through \(10,\) determine whether or not the indicated map \(\phi\) is a homomorphism, and in the cases where \(\phi\) is a homomorphism, determine Kem \(\phi\). $$ \phi: \mathbb{Z} \rightarrow \mathbb{Z}, \text { where } \phi(n)=n-1 $$
Problem 2
All the quotient groups are cyclic and therefore isomorphic to \(\mathbb{Z}_{n}\) for some \(n\). In each case, find this \(n\). $$ \mathbb{Z}_{12} /\langle 8\rangle $$
Problem 2
In Exercises 1 through \(10,\) determine whether or not the indicated map \(\phi\) is a homomorphism, and in the cases where \(\phi\) is a homomorphism, determine Kem \(\phi\). $$ \phi: \mathbb{Z} \rightarrow \mathbb{Z}, \text { where } \phi(n)=3 n $$
Problem 2
Find all the cosets of \(9 \mathbb{Z}\) in \(Z\) and of \(9 \mathbb{Z}\) in \(3 Z\).
Problem 3
All the quotient groups are cyclic and therefore isomorphic to \(\mathbb{Z}_{n}\) for some \(n\). In each case, find this \(n\). $$ \mathbb{Z}_{15} /\langle 6\rangle $$
Problem 3
Determine whether the indicated subgroup is normal in the indicated group. $$ 3 \mathbb{Z} \text { in Z } $$
Problem 3
Let \(G=\langle a\rangle\) be a cyclic group of order \(10 .\) Describe explicitly the elements of \(\operatorname{Aut}(G)\).
