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Section11.3Exercises

1

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

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 \({\mathbb R}\).

3

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

4

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

5

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

6

Describe all of the homomorphisms from \({\mathbb Z}\) to \({\mathbb Z}_{12}\).

7

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

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

  2. List the cosets in \(HN/N\), showing the elements in each coset.

  3. List the cosets in \(H/(H \cap N)\), 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 \(n \in {\mathbb N}\), 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 \(k\), prove that \(H\) is normal in \(G\).

13

Prove or disprove: \({\mathbb Q} / {\mathbb Z} \cong {\mathbb Q}\).

14

Let \(G\) be a finite group and \(N\) a normal subgroup of \(G\). If \(H\) is a subgroup of \(G/N\), prove that \(\phi^{-1}(H)\) is a subgroup in \(G\) of order \(|H| \cdot |N|\), 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 \(\phi(H_1) \subseteq H_2\).

16

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

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 \(\phi(H_1) = H_2\). Prove or disprove that \(G_1/H_1 \cong G_2/H_2\).

18

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

19

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