Math Problem Statement
What is the cyclic permutation notation of (142)^2
Solution
To compute in cyclic permutation notation, we need to apply the permutation twice and see the resulting permutation.
The permutation means:
Step 1: Apply the permutation once.
So, after one application, the result is .
Step 2: Apply the permutation again to the result.
Now, apply once more to the result:
Thus, the result of 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:
This means that applying 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:
- How do you find the inverse of a cyclic permutation?
- What is the order of a permutation, and how do you calculate it?
- How does the composition of two disjoint permutations work?
- Can you explain how to represent a permutation as a product of transpositions?
- 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 , 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