Question

Show that every cyclic permutation \(\left(a_{1} a_{2} . a_{n}\right)\) has the property that for any permutation \(\pi\), $$ \pi\left(a_{1} a_{2} \quad a_{4 t}\right) \pi^{-1} $$ is also a cycle of length \(n\). [Hint It is only necessary to show this for interchanges \(\pi=\left(b_{1} b_{2}\right)\) as cvery permutation is a product of such interchanges ] (a) Show that the conjugacy classes of \(S_{n}\) consist of those permutations having the same cycle structure, eg \((123) \times 45)\) and \((146) \times 23)\) belong to the same corlyugacy class (b) Write ouf all conjugacy classes of \(S_{4}\) and calculate the number of elements in each class

Short answer

Every cyclic permutation retains its length when being conjugated by any permutation in the symmetric group. The conjugacy classes in Sn are formed by permutations sharing the same cycle structure. Therefore, the conjugacy classes in S4 and their sizes are: {1-cycle: 1}, {3-cycle and 1-cycle: 32}, {2 2-cycles: 6}, {4-cycle: 6}, {2-cycle and 2 1-cycle: 18}.

Step-by-step solution

Proving property for cyclic permutations

Let's take the cyclic permutation $\sigma=(a_{1}a_{2}\cdots a_{n})$ and any transposition $\tau=(b_{1}b_{2})$. The result of conjugation $\tau\sigma\tau^{-1}$ is obtained by replacing every element in the cycle $\sigma$ by its image under transposition $\tau$. If $b_{1}$ or $b_{2}$ are not in $\sigma$, $\tau$ does not change the elements in $\sigma$, so $\tau\sigma\tau^{-1}=\sigma$ is also a cycle of length $n$. If $b_{1}$ and $b_{2}$ are both in $\sigma$ and adjacent in the cycle, the transposition just changes their order but not the overall cycle structure, so $\tau\sigma\tau^{-1}$ is still a cycle of length $n$. If $b_{1}$ and $b_{2}$ are both in $\sigma$ but not adjacent in the cycle, the transposition splits the initial cycle into two smaller cycles, however, the reverse transposition $\tau^{-1}$ will merge them back into a single cycle of length $n$. Thus, for any transposition $\tau$, $\tau\sigma\tau^{-1}$ is a cycle of length $n$. Since every permutation is a product of transpositions, the property holds for any permutation.

Understanding conjugacy classes of Sn

Looking back on our previous step, the result of the conjugation operation only depends on the cycle structure of $\sigma$. Two permutations $\sigma$ and $\pi$ that have the same cycle structure indeed form by conjugation the exact same set of cycles, that is, they are in the same conjugacy class. As per the given example, both $(123)(45)$ and $(146)(23)$ have the same structure of having a length-3 and a length-2 cycle.

Calculating conjugacy classes of S4

We can now construct conjugacy classes for $S_{4}$ based on possible cycle structures:\\ 1. Four individual elements, denoted as (1)(2)(3)(4). There is only one permutation belongs to this class, the identity permutation.\\ 2. One cycle of length 3 and one individual element, such as (123)(4). There are 8 different cycles of length 3 and 4 ways to select the remaining element, so a total of 32 elements in this class.\\ 3. Two cycles of length 2, like (12)(34). We can form 3 different cycles of length 2, so the class contains 3x2=6 elements.\\ 4. A cycle of length 4, as in (1234). We can arrange 4 elements in 3!=6 ways, so this class has 6 elements.\\ 5. One cycle of length 2 and two individual elements, such as (12)(3)(4). We can form 6 cycles of length 2 and keep 3 ways to assign the remaining two elements, this class thus has 18 elements. So, the sizes of conjugacy classes in $S_{4}$ are 1, 32, 6, 6, and 18.

Key concepts

Cyclic Permutation

A cyclic permutation is a permutation in which a set of elements is cycled or rotated among themselves while other elements remain unchanged. For example, the permutation \((a_1 a_2 \ldots a_n)\) means that

• \(a_1\) is sent to \(a_2\)
• \(a_2\) is sent to \(a_3\)
• and so forth, with \(a_n\) cycling back to \(a_1\)

Cyclic permutations can include any number of elements, forming a loop that cycles through those specific elements. The beauty of cyclic permutations is their simplicity and ability to be conjugated by any permutation, retaining their cycle structure.

Conjugacy Classes

In the context of permutation groups, conjugacy classes group together permutations that share the same cycle structure. This is important because the group properties depend largely on cycle structure, rather than specific elements within cycles.

For example,

• The permutations \((123)\times(45)\) and \((146)\times(23)\) both consist of a cycle of length 3 and another of length 2.
• As a result, they belong to the same conjugacy class.

Conjugacy classes are a fundamental concept in group theory as they help identify symmetries in structures or systems. The permutations within the same class are considered equivalent under conjugation.

Transpositions

Transpositions are simple but powerful permutations that swap two elements and leave the rest unchanged. They are represented by the notation \((b_1 b_2)\).

You can think of a transposition as a tiny cycle of two elements. Any permutation can be broken down into a series of transpositions. This property is incredibly useful in group theory and allows us to further analyze more complex permutations.

• For example, to perform the permutation \((1 2 3)\), you could use transpositions \((1 2)\), then \((2 3)\).

Understanding transpositions as building blocks helps in manipulating and proving characteristics about more complex permutations.

Sn Groups

The term \(S_n\) refers to the symmetric group on \(n\) elements. These groups consist of all possible permutations of \(n\) distinct elements. They represent the foundation of many concepts in group theory.

• The group \(S_4\) includes all permutations of four elements.
• There are \(4! = 24\) different permutations in \(S_4\).

Each permutation can be expressed through various combinations of cycles and transpositions. The study of \(S_n\) groups is central to exploring symmetry and structure within mathematics, extending to fields like physics and chemistry where symmetrical properties are often crucial.