Introduction to Permutations and Their Orders

The order of a permutation, which is a bijective function of a finite set onto itself, is a fundamental concept in group theory. It is defined as the least common multiple (LCM) of the lengths of the disjoint cycles in the permutation's cycle decomposition. This article aims to explore the maximum possible order of a permutation in the symmetric group (S_{10}).

Understanding Permutation Orders and LCM

Let's begin by clarifying the key concepts: the order of a permutation and the least common multiple (LCM).

The order of a permutation is the smallest positive integer (n) such that applying the permutation (n) times results in the identity permutation. The LCM of a set of numbers is the smallest number that is a multiple of each of the numbers in the set.

Specifically, for permutations, the order of a product of disjoint cycles is the LCM of the individual orders of the cycles.

Maximum Order of a Permutation in (S_{10})

The symmetric group (S_{10}) consists of all possible permutations of 10 elements. We aim to find the maximum possible order of a permutation within this group.

Let's consider the possible cycle structures and their derived orders:

Single 10-cycle: The order is 10. Two 5-cycles: The order is ( text{lcm}(5, 5) 5 ). One 6-cycle and one 4-cycle: The order is ( text{lcm}(6, 4) 12 ). One 7-cycle and one 3-cycle: The order is ( text{lcm}(7, 3) 21 ). One 8-cycle and one 2-cycle: The order is ( text{lcm}(8, 2) 8 ). One 9-cycle and one 1-cycle: The order is ( text{lcm}(9, 1) 9 ). Mixed cycle structures: One 3-cycle, one 3-cycle, and one 4-cycle: The order is ( text{lcm}(3, 3, 4) 12 ). Four 2-cycles: The order is ( text{lcm}(2, 2, 2, 2, 2) 2 ). Three 2-cycles and one 4-cycle: The order is ( text{lcm}(2, 2, 2, 4) 4 ).

From these combinations, the highest order obtained is 21, from one 7-cycle and one 3-cycle, which satisfies ( text{lcm}(7, 3) 21 ).

Verifying the Maximum Order

To ensure that 21 is indeed the maximum order, let's verify that no other combinations can yield a higher order. We'll consider a few more scenarios for illustration:

One 2-cycle, one 2-cycle, and one 2-cycle, with a 4-cycle: The order is ( text{lcm}(2, 2, 2, 4) 4 ). One 2-cycle, one 2-cycle, one 2-cycle, and one 2-cycle: The order is ( text{lcm}(2, 2, 2, 2, 2) 2 ).

These additional cycle structures demonstrate that the maximum possible LCM we can achieve with (S_{10}) is 21. Hence, the maximum order of a permutation in (S_{10}) is 21.

Conclusion

Understanding the maximum order of a permutation in (S_{10}) involves analyzing various cycle structures and their LCMs. By using the least common multiple, we can determine that the highest possible order is 21, achieved by the combination of one 7-cycle and one 3-cycle.

This knowledge is crucial in various mathematical and computational contexts, including solving problems related to group theory and permutation group theory.