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Take A={1,2,3,4,5,6}.Take a partition of A{1,2,3} {4,5,6}. Find a permutation f on A such that the orbits of f are {1,2,3} and {4,5,6}.

Take A={1,2,3,4,5,6}.Take a partition of A{1,2,3} {4,5,6}. Find a permutation f on A such that the orbits of f are {1,2,3} and {4,5,6}.

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Take A={1,2,3,4,5,6}.Take a partition of A{1,2,3} {4,5,6}. Find a permutation f on A such that the orbits of f are {1,2,3} and {4,5,6}.

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Step 1

We are given that A={1,2,3,4,5,6}.

We need to find a permutation f on A such that the orbits of f are {1,2,3} and {4,5,6}.

Note:

We know that the orbit of f ∈ Sn containing j ∈ {1, 2, . . . , n} is the set orbj(f) = {f k(j) : k ∈ Z} ⊆ {1, 2, . . . , n}.

Here, we need to find f on A = S6 such that the orbits of f are {1,2,3} and {4,5,6}.

Which implies that there would be two elements m and n in A such that orbm(f) = {1, 2, 3} and orbn(f) = {4, 5, 6} respectively.

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