Permutation Group S5

In order to understand the maximum possible order of an element in the permutation group S5, let's first define what a permutation group is.

A permutation group is a group that consists of all possible permutations of a set. In this case, the permutation group S5 consists of all possible permutations of the set {1, 2, 3, 4, 5}.

Order of an Element

The order of an element in a group is defined as the smallest positive integer n such that raising the element to the power of n gives the identity element of the group. In other words, it is the smallest positive integer n for which the element repeats itself after n applications.

Finding the Maximum Possible Order

To find the maximum possible order of an element in the permutation group S5, we need to consider the cycle structure of the permutations.

Cycle Structure of Permutations

A cycle is a permutation that moves certain elements to their new positions while leaving the other elements fixed. For example, (123) is a cycle that moves 1 to 2, 2 to 3, and 3 to 1, leaving 4 and 5 fixed.

In the permutation group S5, the maximum possible order of an element is determined by the length of the longest cycle. This is because the order of an element is the least common multiple of the lengths of its disjoint cycles.

Maximum Order in S5

In the permutation group S5, the maximum possible order of an element is 6. This occurs when the element consists of a 2-cycle and a 3-cycle.

For example, consider the element (12)(345) in S5. This element moves 1 to 2, 2 to 1, 3 to 4, 4 to 5, and 5 to 3, leaving no other elements fixed. Applying this element twice will return the set to its original position, making the order of the element 2.

Therefore, the maximum possible order of an element in the permutation group S5 is 6.

Conclusion

In the permutation group S5, the maximum possible order of an element is 6. This occurs when the element consists of a 2-cycle and a 3-cycle. The order of an element is determined by the length of the longest cycle in the element's cycle structure.