Question

Construct a Cayley table for U(12).

Answer 1

Can you clarify what you mean by U(12)? I know of at least two different definitions for that.

Answer 2

I honestly am not sure. This section is on groups if that helps

Answer 3

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.

Answer 4

I'm almost positive that you are correct and that it is the first one

Answer 5

In that case, let's write out the elements of U(12). This is pretty easy to do by hand. We get \(U(12)=\{1,5,7,12\}\).

Answer 6

Now all we do, is draw out a table, and start multiplying elements.

Answer 7

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, \(U(12)=\{1,5,7,11\}\).

Answer 8

oh thank you so much!!!!!!! I'm sorry I haven't replied in a while but that is perfect! Thanks.