Problem 11 - Algebra: Pure and Applied

Question

For \(\tau=(12)(34),\) determine all the elements of the centralizer \(C(\tau)\) of \(\tau\) in \(S_{n}\) for \(n \geq 4\)

Step-by-Step Solution

Answer

Centralizer has 4 elements for 4-cycle part and \((n-4)!\) permutations, giving \(4 \times (n-4)!\) total permutations.

Step 1: Understanding the Centralizer

The centralizer of a permutation \( \tau \) in the symmetric group \( S_n \), denoted \( C(\tau) \), is the set of elements \( g \in S_n \) such that \( g\tau = \tau g \). This means \( C(\tau) \) includes all permutations that commute with \( \tau \).

Step 2: Identify Cycles

The permutation \( \tau = (12)(34) \) is a product of two disjoint 2-cycles. In symmetries, disjoint cycles commute with each other and with any permutation that affects elements outside these cycles.

Step 3: Commuting Permutations

Since \( (12)(34) \) consists of two disjoint 2-cycles, permutations that commute with \( \tau \) can consist of cycles involving \( 1,2,3, \) and \( 4 \). Any permutation within the form \((12)^a(34)^b\) will commute with \( \tau \) where \( a, b \) are 0 or 1.

Step 4: Elements Outside the Cycle

Elements like 5, 6, ..., n for \( n\geq 5 \) (if \( n > 4 \)) can be permuted arbitrarily among themselves while still commuting with \( \tau \), as long as they don't affect the active cycles.

Step 5: Construct the Centralizer

The centralizer \( C(\tau) \) takes the form of permutations such as \((12)^{a}(34)^{b}\sigma \), where \( \sigma \) is any permutation of elements from 5 to \( n \). There are 4 possible combinations for \( (12)^a(34)^b \) because \( a \) and \( b \) can be either 0 or 1, and \((1234)\sigma\) where \( \sigma \in S_{n-4} \).

Step 6: Count Elements

\( (12)^a(34)^b \) gives 4 permutations: \( (), (12), (34), (12)(34) \). For elements 5 to \( n \), there are \( (n-4)! \) permutations in \( S_{n-4} \). Consequently, \( |C(\tau)| = 4 \times (n-4)! \).

Key Concepts

Disjoint Cycles in Permutations

In permutations, a concept that's often encountered is that of disjoint cycles. Disjoint cycles refer to cycles that do not share any common elements. For example, the permutation \( \tau = (12)(34) \) consists of two disjoint 2-cycles. By definition, this means that cycle \((12)\) operates entirely independently of cycle \((34)\).
This independence is a powerful aspect in permutations because it allows us to manipulate cycles separately without affecting each other. Hence, they're crucial when determining commutative permutations in the symmetric group, as disjoint cycles will always commute. Understanding the structure of disjoint cycles helps simplify complex permutation problems, making them much easier to tackle.

Commutative Permutations

Commutative permutations are those that can be applied in any order without changing the outcome. For instance, if two cycles are disjoint, such as \((12)\) and \((34)\) in the permutation \(\tau\), they commute with each other.
This is because the action of one cycle has no impact on the elements in the other. Hence, any permutation formed from powers of \((12)\) and \((34)\), like \((12)^a(34)^b\), where \(a\) and \(b\) can be 0 or 1, will commute with \(\tau\).
Understanding which permutations commute is essential for solving problems related to the centralizer in symmetric groups, simplifying the determination of such a set.

Symmetric Group S_n

The symmetric group \(S_n\) is the set of all possible permutations of \(n\) elements. This group is fundamental in understanding permutation structures and how various elements relate to one another within a group.
No matter how the elements are arranged, each configuration still belongs to \(S_n\), making it a complete collection of all possible permutations. Within \(S_n\), the centralizer of a permutation like \(\tau = (12)(34)\) includes all permutations that commute with \(\tau\).
Comprehending the size and structure of \(S_n\) gives deep insights into possible manipulations of permutations and sets which permutations have the potential to commute based on the cycle structures.

Permutations of Elements Outside Cycles

In scenarios where \(n > 4\), it is possible to permute the elements that don't appear in a cycle. For the centralizer of \(\tau = (12)(34)\), these are elements 5 through \(n\) when \(n \geq 5\).
Such elements can be freely permuted amongst themselves without interfering with the cycles containing elements 1 through 4. This flexibility is critical, as it means an infinite number of combinations can exist for these outside elements while still producing a commutative permutation with \(\tau\).
The permutations of these elements are represented as \(\sigma \) in the expression \((12)^a(34)^b\sigma \), where \(\sigma\) is any permutation of elements 5 to \(n\). Consequently, this flexibility allows a vast array of permutations to form part of the centralizer \(C(\tau)\).