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Section11.3Exercises

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. \({\mathbb Z}_{12}\)

  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. \({\mathbb Q}\)

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.