The Order of an Element in the Group S6

In the group S6, we are given the element (1 2 3)(2 4 5)(4 5 6), and we need to determine its order. 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.

Understanding the Element
To understand the element (1 2 3)(2 4 5)(4 5 6), let's break it down into its cycle notation representation. The element can be written as:
(1 2 3)(2 4 5)(4 5 6)

This means that when we apply this element to a permutation, it will swap the numbers 1 and 2, 2 and 3, 2 and 4, 4 and 5, and 4 and 6. The remaining numbers remain fixed.

Calculating the Order
To find the order of the element, we need to repeatedly apply the element to itself until we obtain the identity element. Let's calculate the powers of the element:

- Applying the element once:
(1 2 3)(2 4 5)(4 5 6) * (1 2 3)(2 4 5)(4 5 6) = (1 2 3)(2 4 5)(4 5 6)

- Applying the element twice:
(1 2 3)(2 4 5)(4 5 6) * (1 2 3)(2 4 5)(4 5 6) * (1 2 3)(2 4 5)(4 5 6) = (1 3 2)(2 5 4)(4 6 5)

- Applying the element thrice:
(1 2 3)(2 4 5)(4 5 6) * (1 2 3)(2 4 5)(4 5 6) * (1 2 3)(2 4 5)(4 5 6) * (1 2 3)(2 4 5)(4 5 6) = (1 2 3)(2 4 5)(4 5 6)

We can observe that applying the element three times results in the same permutation as the identity element. Therefore, the order of the element is 3.

Conclusion
The order of the element (1 2 3)(2 4 5)(4 5 6) in the group S6 is 3.