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Section10.3Exercises

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 $H = 5 {\mathbb Z}$

Hint

(a)

\begin{equation*} \begin{array}{c|cc} & A_4 & (12)A_4 \\ \hline A_4 & A_4 & (12) A_4 \\ (12) A_4 & (12) A_4 & A_4 \end{array} \end{equation*}

(c) $D_4$ is not normal in $S_4\text{.}$

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}, \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}, \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.

Hint

If $a \in G$ is a generator for $G\text{,}$ then $aH$ is a generator for $G/H\text{.}$

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 $G\text{.}$

Hint

For any $g \in G\text{,}$ show that the map $i_g : G \to G$ defined by $i_g : x \mapsto gxg^{-1}$ is an isomorphism of $G$ with itself. Then consider $i_g(H)\text{.}$

12

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

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

Hint

Suppose that $\langle g \rangle$ is normal in $G$ and let $y$ be an arbitrary element of $G\text{.}$ If $x \in C(g)\text{,}$ we must show that $y x y^{-1}$ is also in $C(g)\text{.}$ Show that $(y x y^{-1}) g = g (y x y^{-1})\text{.}$

13

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

\begin{equation*} Z(G) = \{ x \in G : xg = gx \text{ for all } g \in G \}. \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 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 $G\text{.}$

Hint

(a) Let $g \in G$ and $h \in G'\text{.}$ If $h = aba^{-1}b^{-1}\text{,}$ then

\begin{align*} ghg^{-1} & = gaba^{-1}b^{-1}g^{-1}\\ & = (gag^{-1})(gbg^{-1})(ga^{-1}g^{-1})(gb^{-1}g^{-1})\\ & = (gag^{-1})(gbg^{-1})(gag^{-1})^{-1}(gbg^{-1})^{-1}. \end{align*}

We also need to show that if $h = h_1 \cdots h_n$ with $h_i = a_i b_i a_i^{-1} b_i^{-1}\text{,}$ then $ghg^{-1}$ is a product of elements of the same type. However, $ghg^{-1} = g h_1 \cdots h_n g^{-1} = (gh_1g^{-1})(gh_2g^{-1}) \cdots (gh_ng^{-1})\text{.}$