Abstract Algebra: Theory and Applications

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

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

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 $G{L}_2(\mathbb{R})\text{?}$

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 $n\ge 3\text{.}$

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

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

1. Calculate the center of $S_3\text{.}$

2. Calculate the center of $G{L}_2(\mathbb{R})\text{.}$

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 $G^{\prime }=⟨ab{a}^{-1}{b}^{-1}⟩\text{;}$ that is, $G^{\prime }$ is the subgroup of all finite products of elements in $G$ of the form $ab{a}^{-1}{b}^{-1}\text{.}$ The subgroup $G^{\prime }$ is called the commutator subgroup of $G\text{.}$

1. Show that $G^{\prime }$ 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 $G\text{.}$