Determine the order of the permutation in Discrete math I have this question as a homework so please no full answers. I just need help on part b. we learned very briefly about permutation, mainly it's definition and how cycles works but never heard of the identity permutation and the order of the permutation so i actually don't understand what they're asking for. Can anyone help or give an example?
The identity permutation is the permutation that does nothing at all - it sends every number to itself. You could write this in two-row notation (for $n=9$) as
$$\begin{pmatrix} 1&amp; 2&amp; 3 &amp;4&amp; 5 &amp;6 &amp;7&amp;8&amp;9\\ 1&amp; 2 &amp;3 &amp;4 &amp;5&amp; 6&amp;7&amp;8&amp;9\end{pmatrix}$$for instance.

The order is defined in the question - see if you can figure out what it means using the identity here.
a)  The permutation maps 1 into 2 then 3 into 3 then 3 into 1.  (123) is a cycle of order 3. Doing it 3 times maps 123 to 123. The permutation maps 4 into 5 and 5 into 4. (45) is a cycle of order 2. Doing it twice maps 45 to 45.  The permutation maps 6 into 7, 7 into 8, and 8 into 9.  (6789) is a cycle of order 4.  Doing it 4 times maps 6789 to 6789.

b)  The least common multiple of 2, 3, and 4 is 12.  Repeating this permutation twelve times will map 123 to 123 three times but ends back at 123.  Repeating this permutation 12 times will map 45 to 45 6 times but ends back at 45.  Repeating this permutation will map 6789 to 6789 three times but ends back at 6789. 123456789 permuted 12 times winds up back at 123456789.