Math Problem Statement

What is the cyclic permutation notation of (142)^2

Solution

To compute (142)2(142)^2 in cyclic permutation notation, we need to apply the permutation (142)(142) twice and see the resulting permutation.

The permutation (142)(142) means:

  • 141 \rightarrow 4
  • 424 \rightarrow 2
  • 212 \rightarrow 1

Step 1: Apply the permutation (142)(142) once.

  • 141 \rightarrow 4
  • 424 \rightarrow 2
  • 212 \rightarrow 1

So, after one application, the result is (142)(1 \rightarrow 4 \rightarrow 2).

Step 2: Apply the permutation again to the result.

Now, apply (142)(142) once more to the result:

  • 141 \rightarrow 4
  • 424 \rightarrow 2
  • 212 \rightarrow 1

Thus, the result of (142)2(142)^2 is the identity permutation, meaning each element maps back to itself. In cyclic notation, the identity permutation is written as ()(), or it can also be omitted entirely since it represents no change.

Final Answer:

(142)2=()(142)^2 = () This means that applying (142)(142) twice brings everything back to its original position.

Would you like more details or have any questions?

Here are 5 relative questions to expand this concept:

  1. How do you find the inverse of a cyclic permutation?
  2. What is the order of a permutation, and how do you calculate it?
  3. How does the composition of two disjoint permutations work?
  4. Can you explain how to represent a permutation as a product of transpositions?
  5. What happens when you apply a 3-cycle permutation three times?

Tip: The square of a cycle always results in another permutation that maps each element according to the structure of the original cycle. For 3-cycles like (142)(142), the square typically simplifies to the identity permutation.

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Math Problem Analysis

Mathematical Concepts

Permutations
Cyclic Notation
Group Theory

Formulas

(142)^2 = () (identity permutation)

Theorems

Permutation Composition Theorem
Order of a Permutation

Suitable Grade Level

Grades 11-12