Understanding the Maximum Order in Symmetric Group ( S_{12} )

In this article, we delve into the concept of the maximum order of an element within the symmetric group ( S_{12} ). This involves determining the greatest possible order an element can have in this group and then counting the number of such elements. We'll explore the process through cycle decomposition and Landau’s function.

Step 1: Determining the Maximum Order

The order of an element in the symmetric group ( S_n ) is determined by the least common multiple (LCM) of the lengths of the cycles in its cycle decomposition. Our objective is to identify the cycle structure that maximizes the LCM.

Considering Various Cycle Types

One 12-cycle: The least common multiple (LCM) is 12. One 6-cycle and one 6-cycle: The LCM is 6. One 4-cycle, one 4-cycle, and one 2-cycle: The LCM is 4. Four 3-cycles: The LCM is 3. Two 3-cycles and two 2-cycles: The LCM is 6. One 3-cycle, three 2-cycles: The LCM is 3.

From these possibilities, it is clear that the maximum order is achieved by a single 12-cycle. Hence, the maximum order of any element in ( S_{12} ) is 12.

Step 2: Counting Elements of Maximum Order

To count the number of elements with this maximum order, we need to calculate the number of 12-cycles in ( S_{12} ).

The number of ( k )-cycles in ( S_n ) is given by the formula:

[ frac{n!}{k cdot (n-k)!} ]

For ( k 12 ) and ( n 12 ), the number of such cycles is:

[ frac{12!}{12 cdot (12-12)!} frac{12!}{12 cdot 1!} frac{12!}{12} 11! ]

Calculating ( 11! ):

[ 11! 39916800 ]

Therefore, the number of elements in ( S_{12} ) with the maximum order of 12 is 39,916,800.

Landau's Function and Beyond

Landau's function ( g(n) ) provides the maximum order of an element in ( S_n ). In the case of ( S_{12} ), the value of ( g(12) ) is 60, which is achieved by the cycle type 543. Each cycle 543 configuration has an order of 60. The number of such elements is given by:

[ frac{12!}{543} ]

Here, you can experiment by writing the numbers 1-12 in some arbitrary order and using parentheses to create the 543-cycle to see how many times each number is counted.

Conclusion

By understanding the cycle decomposition and applying Landau's function, we can identify and count the elements with the maximum order in the symmetric group ( S_{12} ). This process illuminates the intricate nature of permutation groups and their order properties.