Problem 3
Let \(G=\langle a\rangle\) be a cyclic group of order \(10 .\) Describe explicitly the elements of \(\operatorname{Aut}(G)\).
Problem 4
All the quotient groups are cyclic and therefore isomorphic to \(\mathbb{Z}_{n}\) for some \(n\). In each case, find this \(n\). $$ S_{4} / A_{4} $$
Problem 4
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 \(\phi\). $$ \begin{aligned} \phi: \mathrm{GL}(2, \mathbb{R}) \rightarrow \mathbb{R}^{*}, \text { where } \mathrm{GL}(2, \mathbb{R}) \text { is the general linear group of } 2 \times 2 \text { invertible }\\\ &\text { matrices and } \phi(A)=\operatorname{det} A \end{aligned} $$
Problem 4
Let \(G\) be an Abelian group. Show that the mapping \(\phi: G \rightarrow G\) defined by letting \(\phi(x)=x^{-1}\) for all \(x \in G\) is an automorphism of \(G\).
Problem 5
Find the index of \langle 10\rangle in \(\mathbb{Z}_{12}\).
Problem 5
Determine \(\operatorname{Aut}(\mathbb{Z})\).
Problem 5
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 \(\phi\). $$ \phi: S_{3} \rightarrow \mathbb{Z}_{2}, \text { where } $$ $$ \phi(\sigma)=\left\\{\begin{array}{l} 0 \text { if } \sigma \text { is an even permutation } \\ 1 \text { if } \sigma \text { is an odd permutation } \end{array}\right. $$
Problem 5
All the quotient groups are cyclic and therefore isomorphic to \(\mathbb{Z}_{n}\) for some \(n\). In each case, find this \(n\). $$ D_{4} /\langle\rho\rangle $$
Problem 6
All the quotient groups are cyclic and therefore isomorphic to \(\mathbb{Z}_{n}\) for some \(n\). In each case, find this \(n\). $$ Q_{8} /\langle\mathbf{j}\rangle $$
Problem 6
Show that the mapping \(\phi: S_{3} \rightarrow S_{3}\) defined by letting \(\phi(x)=x^{-1}\) for all \(x \in S_{3}\) is not an automorphism of \(S_{3}\).
