## Exercises10.4Exercises

###### 1.

For each of the following groups $$G\text{,}$$ determine whether $$H$$ is a normal subgroup of $$G\text{.}$$ If $$H$$ is a normal subgroup, write out a Cayley table for the factor group $$G/H\text{.}$$

1. $$G = S_4$$ and $$H = A_4$$

2. $$G = A_5$$ and $$H = \{ (1), (1 \, 2 \, 3), (1 \, 3 \, 2) \}$$

3. $$G = S_4$$ and $$H = D_4$$

4. $$G = Q_8$$ and $$H = \{ 1, -1, I, -I \}$$

5. $$G = {\mathbb Z}$$ and $$H = 5 {\mathbb Z}$$

###### 2.

Find all the subgroups of $$D_4\text{.}$$ Which subgroups are normal? What are all the factor groups of $$D_4$$ up to isomorphism?

###### 3.

Find all the subgroups of the quaternion group, $$Q_8\text{.}$$ Which subgroups are normal? What are all the factor groups of $$Q_8$$ up to isomorphism?

###### 4.

Let $$T$$ be the group of nonsingular upper triangular $$2 \times 2$$ matrices with entries in $${\mathbb R}\text{;}$$ that is, matrices of the form

\begin{equation*} \begin{pmatrix} a & b \\ 0 & c \end{pmatrix}\text{,} \end{equation*}

where $$a\text{,}$$ $$b\text{,}$$ $$c \in {\mathbb R}$$ and $$ac \neq 0\text{.}$$ Let $$U$$ consist of matrices of the form

\begin{equation*} \begin{pmatrix} 1 & x \\ 0 & 1 \end{pmatrix}\text{,} \end{equation*}

where $$x \in {\mathbb R}\text{.}$$

1. Show that $$U$$ is a subgroup of $$T\text{.}$$

2. Prove that $$U$$ is abelian.

3. Prove that $$U$$ is normal in $$T\text{.}$$

4. Show that $$T/U$$ is abelian.

5. Is $$T$$ normal in $$GL_2( {\mathbb R})\text{?}$$

###### 5.

Show that the intersection of two normal subgroups is a normal subgroup.

###### 6.

If $$G$$ is abelian, prove that $$G/H$$ must also be abelian.

###### 7.

Prove or disprove: If $$H$$ is a normal subgroup of $$G$$ such that $$H$$ and $$G/H$$ are abelian, then $$G$$ is abelian.

###### 8.

If $$G$$ is cyclic, prove that $$G/H$$ must also be cyclic.

###### 9.

Prove or disprove: If $$H$$ and $$G/H$$ are cyclic, then $$G$$ is cyclic.

###### 10.

Let $$H$$ be a subgroup of index $$2$$ of a group $$G\text{.}$$ Prove that $$H$$ must be a normal subgroup of $$G\text{.}$$ Conclude that $$S_n$$ is not simple for $$n \geq 3\text{.}$$

###### 11.

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

###### 12.

Define the centralizer of an element $$g$$ in a group $$G$$ to be the set

\begin{equation*} C(g) = \{ x \in G : xg = gx \}\text{.} \end{equation*}

Show that $$C(g)$$ is a subgroup of $$G\text{.}$$ If $$g$$ generates a normal subgroup of $$G\text{,}$$ prove that $$C(g)$$ is normal in $$G\text{.}$$

###### 13.

Recall that the center of a group $$G$$ is the set

\begin{equation*} Z(G) = \{ x \in G : xg = gx \text{ for all } g \in G \}\text{.} \end{equation*}
1. Calculate the center of $$S_3\text{.}$$

2. Calculate the center of $$GL_2 ( {\mathbb R} )\text{.}$$

3. Show that the center of any group $$G$$ is a normal subgroup of $$G\text{.}$$

4. If $$G / Z(G)$$ is cyclic, show that $$G$$ is abelian.

###### 14.

Let $$G$$ be a group and let $$G' = \langle aba^{- 1} b^{-1} \rangle\text{;}$$ that is, $$G'$$ is the subgroup of all finite products of elements in $$G$$ of the form $$aba^{-1}b^{-1}\text{.}$$ The subgroup $$G'$$ is called the commutator subgroup of $$G\text{.}$$

1. Show that $$G'$$ is a normal subgroup of $$G\text{.}$$

2. Let $$N$$ be a normal subgroup of $$G\text{.}$$ Prove that $$G/N$$ is abelian if and only if $$N$$ contains the commutator subgroup of $$G\text{.}$$