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Week 11 Problems

These problems are taken from the Chapter 13 problems in the text.

  1. (Problem 2) List all abelian groups of order $200$ up to isomorphism.

  2. (Problem 6) Let $G$ be an abelian group of order $m$. If $n$ divides $m$, prove that $G$ has a subgroup of order $n$.

  3. (Problem 7). A torsion group is a group where every element has finite order. Prove that if $G$ is an abelian, finitely generated torsion group, then $G$ is finite. Given an example of an infinite abelian torsion group.

  4. (Problem 16). Prove that the dihedral groups $D_{n}$ are solvable for all $n$.

  5. (Problem 18). Let $G$ be a finite abelian group. Show that there is a sequence of subgroups \(G\supset H_{1}\supset H_{2}\supset\cdots\supset H_{k}\supset\{0\}.\) where each of the quotient groups $H_{i}/H_{i+1}$ is cyclic of order $p$ where $p$ is a prime number. Bonus: Prove that any such sequence of subgroups has the same length. What is that length?

Extra

In the notes to this section I claim that the isomorphism classes of abelian groups of order $n$ correspond to sequences $d_1|d_2|\cdots|d_k$ of integers such that $d_1d_2\cdots d_k=n$, with the associated group being the product of the \(\mathbb{Z}_{d_{i}}\). For example, the possible abelian groups of order $36$ correspond to the sequences:

  • \[36\]
  • \[2|18\]
  • \[3|12\]
  • \[6|6\]

Prove this fact.