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## Exercises13.4Exercises

###### 1.

Find all of the abelian groups of order less than or equal to $40$ up to isomorphism.

###### 2.

Find all of the abelian groups of order $200$ up to isomorphism.

###### 3.

Find all of the abelian groups of order $720$ up to isomorphism.

###### 4.

Find all of the composition series for each of the following groups.

1. $\displaystyle {\mathbb Z}_{12}$

2. $\displaystyle {\mathbb Z}_{48}$

3. The quaternions, $Q_8$

4. $\displaystyle D_4$

5. $\displaystyle S_3 \times {\mathbb Z}_4$

6. $\displaystyle S_4$

7. $S_n\text{,}$ $n \geq 5$

8. $\displaystyle {\mathbb Q}$

###### 5.

Show that the infinite direct product $G = {\mathbb Z}_2 \times {\mathbb Z}_2 \times \cdots$ is not finitely generated.

###### 6.

Let $G$ be an abelian group of order $m\text{.}$ If $n$ divides $m\text{,}$ prove that $G$ has a subgroup of order $n\text{.}$

###### 7.

A group $G$ is a torsion group if every element of $G$ has finite order. Prove that a finitely generated abelian torsion group must be finite.

###### 8.

Let $G\text{,}$ $H\text{,}$ and $K$ be finitely generated abelian groups. Show that if $G \times H \cong G \times K\text{,}$ then $H \cong K\text{.}$ Give a counterexample to show that this cannot be true in general.

###### 9.

Let $G$ and $H$ be solvable groups. Show that $G \times H$ is also solvable.

###### 10.

If $G$ has a composition (principal) series and if $N$ is a proper normal subgroup of $G\text{,}$ show there exists a composition (principal) series containing $N\text{.}$

###### 11.

Prove or disprove: Let $N$ be a normal subgroup of $G\text{.}$ If $N$ and $G/N$ have composition series, then $G$ must also have a composition series.

###### 12.

Let $N$ be a normal subgroup of $G\text{.}$ If $N$ and $G/N$ are solvable groups, show that $G$ is also a solvable group.

###### 13.

Prove that $G$ is a solvable group if and only if $G$ has a series of subgroups

\begin{equation*} G = P_n \supset P_{n - 1} \supset \cdots \supset P_1 \supset P_0 = \{ e \} \end{equation*}

where $P_i$ is normal in $P_{i + 1}$ and the order of $P_{i + 1} / P_i$ is prime.

###### 14.

Let $G$ be a solvable group. Prove that any subgroup of $G$ is also solvable.

###### 15.

Let $G$ be a solvable group and $N$ a normal subgroup of $G\text{.}$ Prove that $G/N$ is solvable.

###### 16.

Prove that $D_n$ is solvable for all integers $n\text{.}$

###### 17.

Suppose that $G$ has a composition series. If $N$ is a normal subgroup of $G\text{,}$ show that $N$ and $G/N$ also have composition series.

###### 18.

Let $G$ be a cyclic $p$-group with subgroups $H$ and $K\text{.}$ Prove that either $H$ is contained in $K$ or $K$ is contained in $H\text{.}$

###### 19.

Suppose that $G$ is a solvable group with order $n \geq 2\text{.}$ Show that $G$ contains a normal nontrivial abelian subgroup.

###### 20.

Recall that the commutator subgroup $G'$ of a group $G$ is defined as the subgroup of $G$ generated by elements of the form $a^{-1} b ^{-1} ab$ for $a, b \in G\text{.}$ We can define a series of subgroups of $G$ by $G^{(0)} = G\text{,}$ $G^{(1)} = G'\text{,}$ and $G^{(i + 1)} = (G^{(i)})'\text{.}$

1. Prove that $G^{(i+1)}$ is normal in $(G^{(i)})'\text{.}$ The series of subgroups

\begin{equation*} G^{(0)} = G \supset G^{(1)} \supset G^{(2)} \supset \cdots \end{equation*}

is called the derived series of $G\text{.}$

2. Show that $G$ is solvable if and only if $G^{(n)} = \{ e \}$ for some integer $n\text{.}$

###### 21.

Suppose that $G$ is a solvable group with order $n \geq 2\text{.}$ Show that $G$ contains a normal nontrivial abelian factor group.

###### 22.Zassenhaus Lemma.

Let $H$ and $K$ be subgroups of a group $G\text{.}$ Suppose also that $H^*$ and $K^*$ are normal subgroups of $H$ and $K$ respectively. Then

1. $H^* ( H \cap K^*)$ is a normal subgroup of $H^* ( H \cap K)\text{.}$

2. $K^* ( H^* \cap K)$ is a normal subgroup of $K^* ( H \cap K)\text{.}$

3. $H^* ( H \cap K) / H^* ( H \cap K^*) \cong K^* ( H \cap K) / K^* ( H^* \cap K) \cong (H \cap K) / (H^* \cap K)(H \cap K^*)\text{.}$

###### 23.Schreier's Theorem.

Use the Zassenhaus Lemma to prove that two subnormal (normal) series of a group $G$ have isomorphic refinements.

###### 24.

Use Schreier's Theorem to prove the Jordan-Hölder Theorem.