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Section11.3Exercises

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, \end{equation*}

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

Hint

(a) is a homomorphism with kernel $\{ 1 \}\text{;}$ (c) is not a homomorphism.

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 $\phi\text{.}$

Hint

Since $\phi(m + n) = 7(m+n) = 7m + 7n = \phi(m) + \phi(n)\text{,}$ $\phi$ is a homomorphism.

5

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

Hint

For any homomorphism $\phi : {\mathbb Z}_{24} \rightarrow {\mathbb Z}_{18}\text{,}$ the kernel of $\phi$ must be a subgroup of ${\mathbb Z}_{24}$ and the image of $\phi$ must be a subgroup of ${\mathbb Z}_{18}\text{.}$ Now use the fact that a generator must map to a generator.

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.

Hint

Let $a, b \in G\text{.}$ Then $\phi(a) \phi(b) = \phi(ab) = \phi(ba) = \phi(b)\phi(a)\text{.}$

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 natural 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 $G_1/H_1 \cong G_2/H_2\text{.}$

Hint

Find a counterexample.

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.