The Number of Elements of Order 4 in the Symmetric Group S6

The symmetric group S6 is a topic of great interest in group theory, where the elements' properties are closely studied. One of the intriguing questions is how many elements of order 4 exist within this group. In this article, we will explore and explain the conditions under which an element in S6 can have an order of 4, and we will determine the exact number of such elements.

Understanding Group Elements in S6

In the symmetric group S6, the order of any given element is determined by the least common multiple (LCM) of the sizes of its disjoint cycles in the cycle notation. For example, if an element in S6 can be represented as a product of disjoint cycles, say cycles of lengths a, b, c, etc., then the order of the element is the LCM of a, b, c, etc.

Let's consider the two primary types of elements in S6 that can have an order of 4. These are:

1. Elements of the form (a b c d) (e f), where (a b c d) is a 4-cycle and (e f) is a 2-cycle.

2. Elements of the form (a b c d e f), where all six elements are included in a single 6-cycle.

Calculating the Number of Elements of Order 4 of the First Kind

For the first type of element, we need to form a 4-cycle and a 2-cycle from the 6 elements available. The number of ways to choose 4 elements out of 6 for the 4-cycle is given by the binomial coefficient (binom{6}{4}), and for each choice of 4 elements, the number of ways to arrange them in a 4-cycle is (3!) (since a cycle of length 4 can start at any of its 4 positions and proceed in one of 2 directions, but we divide by 2 to avoid overcounting). Therefore, the number of such 4-cycles is:

[binom{6}{4} times 3! 15 times 6 90]

Similarly, the number of ways to choose 2 elements out of the remaining 2 for the 2-cycle is (binom{2}{2}), and there is only 1 way to arrange 2 elements in a 2-cycle. Thus, the number of such 2-cycles is 1. Therefore, the total number of elements of the first kind is:

[90 times 1 90]

Calculating the Number of Elements of Order 4 of the Second Kind

For the second type of element, which is a single 6-cycle, the number of ways to arrange 6 elements in a 6-cycle is given by the number of permutations of 6 elements divided by the number of ways to arrange the 6-cycle (since a cycle of length 6 can start at any of its 6 positions and proceed in one of 2 directions, but we divide by 2 to avoid overcounting). Therefore, the number of 6-cycles is:

[frac{6!}{6 times 2} frac{720}{12} 60]

However, upon closer inspection, we realize that the 2-cycles in the first part of the calculation must be included in the total count. For the 6-cycle, we need to choose 4 of the 6 elements to form a 4-cycle and the remaining 2 to form a 2-cycle. The number of ways to choose 4 elements out of 6 for the 4-cycle is (binom{6}{4}), and for each choice of 4 elements, the number of ways to arrange them in a 4-cycle is (3!). The number of ways to choose 2 elements out of the remaining 2 for the 2-cycle is (binom{2}{2}), and for each choice, the number of ways to arrange them in a 2-cycle is 1. Therefore, the number of 6-cycles is:

[binom{6}{4} times 3! times 1 15 times 6 90]

Note that this count is the same as the number of elements of the first type, so we do not need to add them separately.

Total Number of Elements of Order 4 in S6

Since the number of elements of both types is the same, the total number of elements of order 4 in S6 is:

[90 90 180]

Furthermore, each element of S6 can be written in a certain form that allows us to calculate the exact number of such elements. The number of 4-cycles that can be formed from 6 elements is:

[360 / 4 90]

Since the remaining 2 elements can only form one 2-cycle, the total number of elements of order 4 is:

[90 times 2 180]

This confirms our previous calculations.

Conclusion

The number of elements of order 4 in the symmetric group S6 is 180. This result is derived from the properties of the elements' disjoint cycles and the combinatorial methods used to count such elements. Understanding the structure and properties of these elements in S6 is crucial for deeper insights into group theory and permutation groups.