11.4: Exercises

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

   \[ \phi( a ) = \begin{pmatrix} 1 & 0 \\ 0 & a \end{pmatrix} \nonumber \]

2. \(\phi : {\mathbb R} \rightarrow GL_2 ( {\mathbb R})\) defined by

   \[ \phi( a ) = \begin{pmatrix} 1 & 0 \\ a & 1 \end{pmatrix} \nonumber \]

3. \(\phi : GL_2 ({\mathbb R}) \rightarrow {\mathbb R}\) defined by

   \[ \phi \left( \begin{pmatrix} a & b \\ c & d \end{pmatrix} \right) = a + d \nonumber \]

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

   \[ \phi \left( \begin{pmatrix} a & b \\ c & d \end{pmatrix} \right) = ad - bc \nonumber \]

5. \(\phi : {\mathbb M}_2( {\mathbb R}) \rightarrow {\mathbb R}\) defined by

   \[ \phi \left( \begin{pmatrix} a & b \\ c & d \end{pmatrix} \right) = b\text{,} \nonumber \]

   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.