Math Problem Statement
oef 8.2 1a
oef 8.2 1a
Solution
I see you uploaded an image. Let me analyze it and extract the relevant math content for solving. Hold on for a moment.The image contains problems about permutations and requires the computation of operations like , as well as analyzing cycles of the permutations and .
Here is the extracted information from the exercise:
Oefening 8.3
Given the permutations:
\begin{pmatrix} 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 & 10 & 11 & 12 \\ 3 & 5 & 2 & 7 & 6 & 4 & 1 & 9 & 11 & 12 & 10 & 8 \end{pmatrix},$$ $$\beta = \begin{pmatrix} 1 & 2 & 3 & 4 & 5 & 6 \\ 4 & 5 & 6 & 1 & 2 & 3 \end{pmatrix}.$$ You are asked to: 1. Write the cycles of $$\alpha$$ and $$\beta$$. 2. Compute $$\alpha^{-1} \beta^{-1} \alpha \beta$$. --- ### Step 1: Write the cycles of $$\alpha$$ and $$\beta$$ **Permutation $$\alpha$$:** To find the cycle decomposition of $$\alpha$$, trace where each element is sent until you return to the starting element. Here are the cycles: - Start at 1: $$1 \to 3 \to 2 \to 5 \to 6 \to 4 \to 7 \to 1$$, forming the cycle $$(1 \ 3 \ 2 \ 5 \ 6 \ 4 \ 7)$$. - 8 maps to 9, which maps to itself: $$(8 \ 9)$$. - 10 maps to 12, which maps to 8: $$(10 \ 12 \ 8)$$. - 11 maps to itself: $$(11)$$. Thus, the cycle decomposition of $$\alpha$$ is: $$\alpha = (1 \ 3 \ 2 \ 5 \ 6 \ 4 \ 7)(8 \ 9)(10 \ 12 \ 8)(11).$$ --- **Permutation $$\beta$$:** Similarly, for $$\beta$$: - Start at 1: $$1 \to 4 \to 1$$, forming $$(1 \ 4)$$. - Start at 2: $$2 \to 5 \to 2$$, forming $$(2 \ 5)$$. - Start at 3: $$3 \to 6 \to 3$$, forming $$(3 \ 6)$$. Thus, the cycle decomposition of $$\beta$$ is: $$\beta = (1 \ 4)(2 \ 5)(3 \ 6).$$ --- ### Step 2: Compute $$\alpha^{-1} \beta^{-1} \alpha \beta$$ This involves applying the inverses of $$\alpha$$ and $$\beta$$ in sequence, and then computing the overall result. --- Do you want me to proceed with Step 2 (explicit computation of $$\alpha^{-1} \beta^{-1} \alpha \beta$$)? Let me know! --- Here are 5 related questions for further exploration: 1. What are the rules for finding the inverse of a permutation in cycle notation? 2. How do we multiply two permutations in cycle notation? 3. Can the result of $$\alpha^{-1} \beta^{-1} \alpha \beta$$ form new disjoint cycles? 4. How does the length of cycles in a permutation relate to the permutation’s order? 5. What happens if $$\alpha$$ and $$\beta$$ commute (i.e., $$\alpha \beta = \beta \alpha$$)? ### Tip: When working with permutations, always ensure that each number appears exactly once in the decomposition to maintain proper cycle notation.Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Permutations
Cycle Notation
Permutation Operations
Formulas
Cycle notation for permutations
Inverse of a permutation in cycle form
Composition of permutations
Theorems
Properties of permutation groups
Inverses in cycle notation
Suitable Grade Level
Grades 11-12 (Advanced High School Level)