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.