13.4: Exercises

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

\[ G = P_n \supset P_{n - 1} \supset \cdots \supset P_1 \supset P_0 = \{ e \} \nonumber \]

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

   \[ G^{(0)} = G \supset G^{(1)} \supset G^{(2)} \supset \cdots \nonumber \]

   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.