Number of Elements of Order 4 in Alternating Group A4

Introduction
The alternating group A4 is a subgroup of the symmetric group S4, consisting of all even permutations of four elements. In this response, we will determine the number of elements of order 4 in the alternating group A4.

Order of an Element
The order of an element in a group is the smallest positive integer n such that raising the element to the power of n yields the identity element. In other words, the order of an element is the smallest positive integer n for which a^n = e, where a is the element and e is the identity element.

Determining Elements of Order 4
To find the elements of order 4 in A4, we need to consider the possible cycle structures of permutations in A4. A permutation with a cycle structure of length 4 or two cycles of length 2 has order 4.

Cycle Structure of Permutations in A4
In A4, the possible cycle structures are:
- 4-cycles: Permutations with a single cycle of length 4
- 2-cycles: Permutations with two cycles of length 2

Counting 4-cycles
To count the number of 4-cycles in A4, we can use the formula for the number of permutations with a given cycle structure. The formula is given by:

N(n1, n2, ..., nk) = (4!/(1!^n1 * 2!^n2 * ... * k!^nk)) * (1^n1 * 2^n2 * ... * k^nk)

where n1, n2, ..., nk are the lengths of the cycles.

In our case, we have one 4-cycle, so the formula becomes:

N(1) = (4!/(1!^1)) * (1^1) = (4 * 3 * 2 * 1)/(1) = 24/1 = 24

Therefore, there are 24 4-cycles in A4.

Counting 2-cycles
To count the number of 2-cycles in A4, we again use the formula for the number of permutations with a given cycle structure. In this case, we have two cycles of length 2, so the formula becomes:

N(2, 2) = (4!/(1!^2 * 2!^2)) * (1^2 * 2^2) = (4 * 3 * 2 * 1)/(1 * 2 * 2) = 24/4 = 6

Therefore, there are 6 2-cycles in A4.

Total Number of Elements of Order 4
To determine the total number of elements of order 4 in A4, we sum the number of 4-cycles and the number of 2-cycles:

Total = 24 (4-cycles) + 6 (2-cycles) = 30

Therefore, there are 30 elements of order 4 in the alternating group A4.

Conclusion
In conclusion, the alternating group A4 contains 30 elements of order 4. This is determined by considering the possible cycle structures of permutations