Chapter 2

Determine whether the indicated subgroup is normal in the indicated group. $$ A_{3} \text { in } S_{3} $$

All the quotient groups are cyclic and therefore isomorphic to \(\mathbb{Z}_{n}\) for some \(n\). In each case, find this \(n\). $$ \mathbb{Z}_{6} /\langle 2\rangle $$

Find all the cosets of the subgroup \(5 Z\) in \(\mathbb{Z}\).

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 \(\phi\). $$ \phi: \mathbb{Z} \rightarrow \mathbb{Z}, \text { where } \phi(n)=n-1 $$

All the quotient groups are cyclic and therefore isomorphic to \(\mathbb{Z}_{n}\) for some \(n\). In each case, find this \(n\). $$ \mathbb{Z}_{12} /\langle 8\rangle $$

Find all the cosets of \(9 \mathbb{Z}\) in \(Z\) and of \(9 \mathbb{Z}\) in \(3 Z\).

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 \(\phi\). $$ \phi: \mathbb{Z} \rightarrow \mathbb{Z}, \text { where } \phi(n)=3 n $$

Determine whether the indicated subgroup is normal in the indicated group. $$ 3 \mathbb{Z} \text { in Z } $$

All the quotient groups are cyclic and therefore isomorphic to \(\mathbb{Z}_{n}\) for some \(n\). In each case, find this \(n\). $$ \mathbb{Z}_{15} /\langle 6\rangle $$

Let \(G=\langle a\rangle\) be a cyclic group of order \(10 .\) Describe explicitly the elements of \(\operatorname{Aut}(G)\).

Find all the cosets of \langle 6\rangle in \(Z_{12}\) and all the cosets of \langle 6\rangle in the subgroup \langle 2\rangle of \(\mathbb{Z}_{12}\).

All the quotient groups are cyclic and therefore isomorphic to \(\mathbb{Z}_{n}\) for some \(n\). In each case, find this \(n\). $$ S_{4} / A_{4} $$

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 \(G\).

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 \(\phi\). $$ \begin{aligned} \phi: \mathrm{GL}(2, \mathbb{R}) \rightarrow \mathbb{R}^{*}, \text { where } \mathrm{GL}(2, \mathbb{R}) \text { is the general linear group of } 2 \times 2 \text { invertible }\\\ &\text { matrices and } \phi(A)=\operatorname{det} A \end{aligned} $$

All the quotient groups are cyclic and therefore isomorphic to \(\mathbb{Z}_{n}\) for some \(n\). In each case, find this \(n\). $$ D_{4} /\langle\rho\rangle $$

Determine \(\operatorname{Aut}(\mathbb{Z})\).

Find the index of \langle 10\rangle in \(\mathbb{Z}_{12}\).

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 \(\phi\). $$ \phi: S_{3} \rightarrow \mathbb{Z}_{2}, \text { where } $$ $$ \phi(\sigma)=\left\\{\begin{array}{l} 0 \text { if } \sigma \text { is an even permutation } \\ 1 \text { if } \sigma \text { is an odd permutation } \end{array}\right. $$

Determine whether the indicated subgroup is normal in the indicated group. $$ \\{\pm 1, \pm \mathbf{j}\\} \text { in } Q_{8} $$

All the quotient groups are cyclic and therefore isomorphic to \(\mathbb{Z}_{n}\) for some \(n\). In each case, find this \(n\). $$ Q_{8} /\langle\mathbf{j}\rangle $$

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 \(S_{3}\).

Determine whether the indicated subgroup is normal in the indicated group. $$ K=\left\\{\rho_{0},(12)(34),(13)(24),(14)(23)\right\\} \text { in } S_{4} $$

Find the order of the indicated element in the indicated quotient group. $$ 3+\langle 8\rangle \text { in } \mathbb{Z}_{12} /\langle 8\rangle $$

Let \(G\) be a group, \(H \triangleleft G, \phi \in \operatorname{Aut}(G)\). Show that \(\phi(H) \triangleleft G\).

Determine whether the indicated subgroup is normal in the indicated group. $$ \langle(123)\rangle \text { in } S_{4} $$

Find the order of the indicated element in the indicated quotient group. $$ 3+\langle 6\rangle \text { in } \mathbb{Z}_{15} /\langle 6\rangle $$

Let \(H=\left\\{\phi \in S_{n} \mid \phi(n)=n\right\\}\). Find the index of \(H\) in \(S_{n}\).

Find the order of the indicated element in the indicated quotient group. $$ 2+\langle 6\rangle \text { in } \mathbb{Z}_{15} /\langle 6\rangle $$

Show that \(\operatorname{Inn}\left(S_{3}\right) \cong S_{3}\).

Let \(H\) be a subgroup of a group \(G\). For any \(a, b \in G,\) let \(a \sim b\) if and only if \(a b^{-1} \in H\). Show that the relation \(\sim\) so defined is an equivalence relation on \(G,\) with equivalence classes the right cosets \(\mathrm{Ha}\) of \(\mathrm{H}\).

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 \(\phi\). $$ \text { \phi: } \mathbb{Z}_{6} \rightarrow \mathbb{Z}_{2} \text { , where } \phi(x)=\text { the remainder of } x \text { mod } 2 $$

For \(p\) a prime show that \(\operatorname{Aut}\left(\mathbb{Z}_{p}\right) \cong \mathbb{Z}_{p-1}\).

Let \(H\) be a subgroup of a group \(G\). Show that for any \(a \in G\) we have \(|H a|=|H|\).

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 \(\phi\). $$ \phi: \mathbb{Z}_{7} \rightarrow \mathbb{Z}_{2}, \text { where } \phi(x)=\text { the remainder of } x \text { mod } 2 $$

Determine whether the indicated subgroup is normal in the indicated group. Find all the normal subgroups in \(\mathrm{GL}\left(2, \mathbb{Z}_{2}\right)\), the general linear group of \(2 \times 2\) matrices with entries from \(\mathbb{Z}_{2}\).

Let \(Q_{8}\) be the quarternion group. Show that \(\operatorname{Inn}\left(Q_{8}\right)=V,\) the Klein 4 -group.

Determine whether the indicated subgroup is normal in the indicated group. For \(r \in \mathbb{R}^{*}\) let \(r I=\left[\begin{array}{rr}r & 0 \\ 0 & r\end{array}\right]\). Show that \(H=\left\\{r l \mid r \in \mathbb{R}^{*}\right\\}\) is a normal subgroup of \(\mathrm{GL}(2, \mathbb{R})\)

Show that \(\mid\) Aut \(\left(D_{4}\right) \mid \leq 8\).

Let \(H\) be a subgroup of a group \(G\). Show for any \(a \in G\) that \(a H=H\) if and only if \(a \in H .\)

For \(V\) the Klein 4 -group show that \(\operatorname{Aut}(V) \cong \mathrm{GL}\left(2, \mathbb{Z}_{2}\right),\) the general linear group of \(2 \times 2\) matrices with entries from \(\mathbb{Z}_{2}\).

Let \(H=5 \mathbb{Z}\) in \(\mathbb{Z}\). Determine whether the following cosets of \(H\) are the same: (a) \(12+H\) and \(27+H\) (b) \(13+H\) and \(-2+H\) (c) \(126+H\) and \(-1+H\)

A subgroup \(H\) of a group \(G\) is called a characteristic subgroup of \(G\) if for all \(\phi \in \operatorname{Aut}(G)\) we have \(\phi(H)=H\). Show that if \(H\) is a characteristic subgroup of \(G,\) then \(H \triangleleft G\).

Determine whether the indicated subgroup is normal in the indicated group. Show that if \(H \triangleleft G\) and \(K \triangleleft G\), then \(H \cap K \triangleleft G\).

Find all possible nontrivial homomorphisms between the indicated groups. $$ \phi: D_{4} \rightarrow \mathbb{Z}_{4} $$

A subgroup \(H\) of a group \(G\) is called a characteristic subgroup of \(G\) if for all \(\phi \in \operatorname{Aut}(G)\) we have \(\phi(H)=H\). Show that if \(H\) is the only subgroup of \(G\) of order \(n\), then \(H\) is a characteristic subgroup of \(G\).

Let \(G=\langle a\rangle\) be a cyclic group of order 60 , and \(H=\left\langle a^{35}\right\rangle\). List all the left cosets of \(H\) in \(G\).

Find all possible homomorphisms from \(\mathbb{Z}\) to \(\mathbb{Z}\).

Determine whether the indicated subgroup is normal in the indicated group. Let \(H\) be a subgroup of a group \(G,\) and suppose that for every \(x \in G\) there is a \(y \in G\) such that \(x H=H y\). Show that \(H \triangleleft G\).

Find all possible nontrivial homomorphisms between the indicated groups. $$ \phi: \mathbb{Z}_{5} \rightarrow \mathbb{Z}_{10} $$

A subgroup \(H\) of a group \(G\) is called a characteristic subgroup of \(G\) if for all \(\phi \in \operatorname{Aut}(G)\) we have \(\phi(H)=H\). Let \(G\) be a group, \(H\) a normal subgroup of \(G,\) and \(K\) a characteristic subgroup of \(H .\) Show that \(K\) is a normal subgroup of \(G\).