Math Problem Statement
Let 𝜎 = ( 1 2 3 4 5 6 7 8 9 9 7 8 3 2 6 5 4 1) be a permutation in 𝑆9 .
- Write 𝜎 as a product of disjoint cycles
Determine the order
Solution
To write the permutation as a product of disjoint cycles and determine its order, we will follow these steps:
- Identify the cycles by tracking where each element maps.
- Write in cycle notation.
- Determine the order of by finding the least common multiple (LCM) of the lengths of the disjoint cycles.
Given permutation :
1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 \\ 9 & 7 & 8 & 3 & 2 & 6 & 5 & 4 & 1 \end{pmatrix}$$ ### Step 1: Identify the Cycles Start with 1: $$1 \rightarrow 9 \rightarrow 1$$ This forms the cycle $$(1 \ 9)$$. Move to the next smallest number not yet included in a cycle, which is 2: $$2 \rightarrow 7 \rightarrow 5 \rightarrow 2$$ This forms the cycle $$(2 \ 7 \ 5)$$. Move to the next smallest number not yet included in a cycle, which is 3: $$3 \rightarrow 8 \rightarrow 4 \rightarrow 3$$ This forms the cycle $$(3 \ 8 \ 4)$$. Number 6 maps to itself: $$6 \rightarrow 6$$ This forms the cycle $$(6)$$, but we typically omit single element cycles in notation. ### Step 2: Write $$\sigma$$ in Cycle Notation Combining all the disjoint cycles, we get: $$\sigma = (1 \ 9)(2 \ 7 \ 5)(3 \ 8 \ 4)$$ ### Step 3: Determine the Order of $$\sigma$$ The order of a permutation is the LCM of the lengths of its disjoint cycles. The lengths of the cycles are: - $$(1 \ 9)$$: length 2 - $$(2 \ 7 \ 5)$$: length 3 - $$(3 \ 8 \ 4)$$: length 3 Calculate the LCM of 2, 3, and 3: $$\text{LCM}(2, 3, 3) = 6$$ Therefore, the order of $$\sigma$$ is 6. ### Summary - The permutation $$\sigma$$ can be written as the product of disjoint cycles: $$(1 \ 9)(2 \ 7 \ 5)(3 \ 8 \ 4)$$. - The order of $$\sigma$$ is 6. Do you want more details or have any questions? Here are some related questions to expand your understanding: 1. How do you find the cycle structure of a permutation in $$S_n$$? 2. What is the significance of disjoint cycles in permutation groups? 3. How do you calculate the least common multiple (LCM) of multiple numbers? 4. What are the properties of the order of a permutation? 5. How do single element cycles affect the cycle notation? 6. What is the symmetric group $$S_n$$ and its properties? 7. How do permutations relate to matrix representations? 8. How can you determine if two permutations are conjugate in $$S_n$$? **Tip:** When working with permutations, always ensure that you check the mapping of each element carefully to avoid mistakes in identifying cycles.Ask a new question for Free
By Image
Drop file here or Click Here to upload
Math Problem Analysis
Mathematical Concepts
Permutations
Cycle notation
Order of permutations
Formulas
LCM (Least Common Multiple)
Theorems
-
Suitable Grade Level
Advanced High School
Related Recommendation
Find the Order of Permutation σ = (1 9)(2 7 5)(3 8 4)
Computation of Powers and Products of Permutations in S6
Analysis of Permutation σ in S7 with Disjoint Cycles
Write Permutation σ as Product of Disjoint Cycles: Detailed Solution
Decomposing Permutation σ = (24)(1759)(428)(735) into Disjoint Cycles and Determining Parity