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On Monday last week, one of the problems we looked at was the following.

Problem 104 (A Direct Product Of Cyclic Groups Is Cyclic If And Only If The Groups Have Relatively Prime Orders)

Suppose that $G$ and $H$ are finite cyclic groups. Prove that $G\oplus H$ is cyclic if and only if $|G|$ and $|H|$ are relatively prime.

Your assignment for Monday is to extent this problem to work with the direct product of a finite number of cyclic groups.

Exercise (A Direct Product Of Cyclic Groups Is Cyclic If And Only If The Groups Have Relatively Prime Orders)

Use induction to extend the previous result to the external direct product of $n$ cyclic groups. So prove that if $G_1,G_2,\ldots,G_n$ are $n$ cyclic groups, then $G_1\oplus G_2\oplus \cdots\oplus G_n$ is cyclic if and only if $|G_i|$ and $|G_j|$ are relatively prime when $i\neq j$.

Click to see a hint.

This exercise require that you use induction on an "if $p$ then $q$" statement. In addition, the $q$ part of the statement involves an if and only if. You must be very careful how you tackle this problem.

Problem.Induction Mastery Two

Solve the previous exercise. Pay close attention to your logical order. What are you assuming at each stage? Be very careful with the order in which you place your assumptions, statements, and conclusions. This problem is extremely easy to mess up on.

Once you have a solution, your assignment is to type up your solution using a $\LaTeX$ processor that you have install on your computer, or used on a campus computer (there are a few in the computer lab that have LaTeX). Use this template (latex-template.tex) to type your file.

Here are some instructions for obtaining LaTeX.

  1. If you use Windows, Download MikTeX from http://miktex.org/download and install MikTeX.
    • This will give your compute the ability to build PDF documents from $\LaTeX$ files without needing access to the internet.
    • This will also install a program called TeXWorks where you edit your $\LaTeX$ code (generally saved in a .tex file).
  2. If you use Mac, then download MacTeX from http://tug.org/mactex/. There should be an option to install TeXWorks as part of your installation. Do this.
  3. If you use Linux (Caley), then I followed the information on http://askubuntu.com/questions/163682/how-do-i-install-the-latest-tex-live-2012 to install TeX. You'll then need an editor (like TeXWorks, see http://www.tug.org/texworks/).

Once you have a working copy of LaTeX, you should be able to download the following template (latex-template.tex), double click on the download to open TeXWorks, and then type your solution and build your PDF file.

For more problems, see AllProblems