## Exercises11.4Exercises

###### 1.

Prove that $$\det( AB) = \det(A) \det(B)$$ for $$A, B \in GL_2( {\mathbb R} )\text{.}$$ This shows that the determinant is a homomorphism from $$GL_2( {\mathbb R} )$$ to $${\mathbb R}^*\text{.}$$

###### 2.

Which of the following maps are homomorphisms? If the map is a homomorphism, what is the kernel?

1. $$\phi : {\mathbb R}^\ast \rightarrow GL_2 ( {\mathbb R})$$ defined by

\begin{equation*} \phi( a ) = \begin{pmatrix} 1 & 0 \\ 0 & a \end{pmatrix} \end{equation*}
2. $$\phi : {\mathbb R} \rightarrow GL_2 ( {\mathbb R})$$ defined by

\begin{equation*} \phi( a ) = \begin{pmatrix} 1 & 0 \\ a & 1 \end{pmatrix} \end{equation*}
3. $$\phi : GL_2 ({\mathbb R}) \rightarrow {\mathbb R}$$ defined by

\begin{equation*} \phi \left( \begin{pmatrix} a & b \\ c & d \end{pmatrix} \right) = a + d \end{equation*}
4. $$\phi : GL_2 ( {\mathbb R}) \rightarrow {\mathbb R}^\ast$$ defined by

\begin{equation*} \phi \left( \begin{pmatrix} a & b \\ c & d \end{pmatrix} \right) = ad - bc \end{equation*}
5. $$\phi : {\mathbb M}_2( {\mathbb R}) \rightarrow {\mathbb R}$$ defined by

\begin{equation*} \phi \left( \begin{pmatrix} a & b \\ c & d \end{pmatrix} \right) = b\text{,} \end{equation*}

where $${\mathbb M}_2( {\mathbb R})$$ is the additive group of $$2 \times 2$$ matrices with entries in $${\mathbb R}\text{.}$$

###### 3.

Let $$A$$ be an $$m \times n$$ matrix. Show that matrix multiplication, $$x \mapsto Ax\text{,}$$ defines a homomorphism $$\phi : {\mathbb R}^n \rightarrow {\mathbb R}^m\text{.}$$

###### 4.

Let $$\phi : {\mathbb Z} \rightarrow {\mathbb Z}$$ be given by $$\phi(n) = 7n\text{.}$$ Prove that $$\phi$$ is a group homomorphism. Find the kernel and the image of $$\phi\text{.}$$

###### 5.

Describe all of the homomorphisms from $${\mathbb Z}_{24}$$ to $${\mathbb Z}_{18}\text{.}$$

###### 6.

Describe all of the homomorphisms from $${\mathbb Z}$$ to $${\mathbb Z}_{12}\text{.}$$

###### 7.

In the group $${\mathbb Z}_{24}\text{,}$$ let $$H = \langle 4 \rangle$$ and $$N = \langle 6 \rangle\text{.}$$

1. List the elements in $$HN$$ (we usually write $$H + N$$ for these additive groups) and $$H \cap N\text{.}$$

2. List the cosets in $$HN/N\text{,}$$ showing the elements in each coset.

3. List the cosets in $$H/(H \cap N)\text{,}$$ showing the elements in each coset.

4. Give the correspondence between $$HN/N$$ and $$H/(H \cap N)$$ described in the proof of the Second Isomorphism Theorem.

###### 8.

If $$G$$ is an abelian group and $$n \in {\mathbb N}\text{,}$$ show that $$\phi : G \rightarrow G$$ defined by $$g \mapsto g^n$$ is a group homomorphism.

###### 9.

If $$\phi : G \rightarrow H$$ is a group homomorphism and $$G$$ is abelian, prove that $$\phi(G)$$ is also abelian.

###### 10.

If $$\phi : G \rightarrow H$$ is a group homomorphism and $$G$$ is cyclic, prove that $$\phi(G)$$ is also cyclic.

###### 11.

Show that a homomorphism defined on a cyclic group is completely determined by its action on the generator of the group.

###### 12.

If a group $$G$$ has exactly one subgroup $$H$$ of order $$k\text{,}$$ prove that $$H$$ is normal in $$G\text{.}$$

###### 13.

Prove or disprove: $${\mathbb Q} / {\mathbb Z} \cong {\mathbb Q}\text{.}$$

###### 14.

Let $$G$$ be a finite group and $$N$$ a normal subgroup of $$G\text{.}$$ If $$H$$ is a subgroup of $$G/N\text{,}$$ prove that $$\phi^{-1}(H)$$ is a subgroup in $$G$$ of order $$|H| \cdot |N|\text{,}$$ where $$\phi : G \rightarrow G/N$$ is the canonical homomorphism.

###### 15.

Let $$G_1$$ and $$G_2$$ be groups, and let $$H_1$$ and $$H_2$$ be normal subgroups of $$G_1$$ and $$G_2$$ respectively. Let $$\phi : G_1 \rightarrow G_2$$ be a homomorphism. Show that $$\phi$$ induces a homomorphism $$\overline{\phi} : (G_1/H_1) \rightarrow (G_2/H_2)$$ if $$\phi(H_1) \subset H_2\text{.}$$

###### 16.

If $$H$$ and $$K$$ are normal subgroups of $$G$$ and $$H \cap K = \{ e \}\text{,}$$ prove that $$G$$ is isomorphic to a subgroup of $$G/H \times G/K\text{.}$$

###### 17.

Let $$\phi : G_1 \rightarrow G_2$$ be a surjective group homomorphism. Let $$H_1$$ be a normal subgroup of $$G_1$$ and suppose that $$\phi(H_1) = H_2\text{.}$$ Prove or disprove that $$G_1/H_1 \cong G_2/H_2\text{.}$$

###### 18.

Let $$\phi : G \rightarrow H$$ be a group homomorphism. Show that $$\phi$$ is one-to-one if and only if $$\phi^{-1}(e) = \{ e \}\text{.}$$

###### 19.

Given a homomorphism $$\phi :G \rightarrow H$$ define a relation $$\sim$$ on $$G$$ by $$a \sim b$$ if $$\phi(a) = \phi(b)$$ for $$a, b \in G\text{.}$$ Show this relation is an equivalence relation and describe the equivalence classes.