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Section13.3Exercises

¶
1

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

Hint

There are three possible groups.

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

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

  3. The quaternions, $Q_8$

  4. $D_4$

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

  6. $S_4$

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

  8. ${\mathbb Q}$

Hint

(a) $\{ 0 \} \subset \langle 6 \rangle \subset \langle 3 \rangle \subset {\mathbb Z}_{12}\text{;}$ (e) $\{ (1) \} \times \{ 0 \} \subset \{ (1), (123), (132) \} \times \{ 0 \} \subset S_3 \times \{ 0 \} \subset S_3 \times \langle 2 \rangle\subset S_3 \times {\mathbb Z}_4\text{.}$

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

Hint

Use the Fundamental Theorem of Finitely Generated Abelian Groups.

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.

Hint

If $N$ and $G/N$ are solvable, then they have solvable series

\begin{gather*} N = N_n \supset N_{n - 1} \supset \cdots \supset N_1 \supset N_0 = \{ e \}\\ G/N = G_n/N \supset G_{n - 1}/N \supset \cdots G_1/N \supset G_0/N = \{ N \}. \end{gather*}
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 $n\text{.}$

Hint

Use the fact that $D_n$ has a cyclic subgroup of index 2.

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 $G'$ 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 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.

Hint

$G/G'$ is abelian.

22Zassenhaus 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. $H^* ( H \cap K) / H^* ( H \cap K^*) \cong K^* ( H \cap K) / K^* ( H^* \cap K) \cong (H \cap K) / (H^* \cap K)(H \cap K^*)\text{.}$

23Schreier'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.