Could you explain how to show this in easiest possible way (in detail)? 6.26. Let n > 2. Show that every element of S, can be written as a product oftranspositions of the form (1 i), for various i.

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Description: Could you explain how to show this in easiest possible way (in detail)? 6.26. Let n > 2. Show that every element of S, can be written as a product oftranspositions of the form (1 i), for various i.

Answered: 6.26. Let n > 2. Show that every…

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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Could you explain how to show this in easiest possible way (in detail)?

6.26. Let n > 2. Show that every element of S, can be written as a product of
transpositions of the form (1 i), for various i.

Expert Solution

Step 1

TO PROVE:

Every element of $S_{n}$ can be written as a product of transpositions.

PROOF:

By a theorem of permutation cycles, it is clear that

Every permutation in $S_{n}$ can be written as the product of disjoint cycles.

And every cycle of at least two length can be written as the product of 3 or 1.

So, any element in $S_{n}$ is of the disjoint cycle from like $(1 2 3 4 5 . . . . . . n)$ which can be written as $(1 2) (1 3) (1 4) . . . . . . . (1 n)$ .

Hence, it can written as a product of transpositions.

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