anonymous
  • anonymous
Construct a Cayley table for U(12).
Mathematics
  • Stacey Warren - Expert brainly.com
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chestercat
  • chestercat
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KingGeorge
  • KingGeorge
Can you clarify what you mean by U(12)? I know of at least two different definitions for that.
anonymous
  • anonymous
I honestly am not sure. This section is on groups if that helps
KingGeorge
  • KingGeorge
Well U(n) is used as notation for two completely different groups. One group is the set of natural number less than 12, and coprime to 12. The other group is a specific matrix group. Does either of these sound familiar? If I had to take a guess, it would be the first one because you can actually write a cayley table for it. The second group would have a very very large table.

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anonymous
  • anonymous
I'm almost positive that you are correct and that it is the first one
KingGeorge
  • KingGeorge
In that case, let's write out the elements of U(12). This is pretty easy to do by hand. We get \(U(12)=\{1,5,7,12\}\).
KingGeorge
  • KingGeorge
Now all we do, is draw out a table, and start multiplying elements.
KingGeorge
  • KingGeorge
This is just a standard multiplication table, except that it's done modulo 12.\[\begin{array}{|r|c|c|c|c|}\hline \times&1&5&7&11\\\hline 1&1&5&7&11\\\hline 5&5&1&11&7\\\hline 7&7&11&1&5\\\hline 11&11&7&5&1\\\hline \end{array}\]And I made a mistake above, \(U(12)=\{1,5,7,11\}\).
anonymous
  • anonymous
oh thank you so much!!!!!!! I'm sorry I haven't replied in a while but that is perfect! Thanks.

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