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## 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 $H = \{ (1), (123), (132) \}$

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 $G\text{.}$