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Prove that for any k ? 2, a k–cycle can be expressed as a composition of...

Prove that for any k ? 2, a k–cycle can be expressed as a composition of exactly k ? 1 2–cycles.

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Answer #1

We apply induction on k

For k=2

which is 1'' 2-cycle

Now Assume result hold for each n

Now consider

Where so by Induction hypothesis can be break down in (k-2), 2-cycles

Henceforth can be break down in to (k-1) 2-cycles which proves our assertion

Note breaking of is like

For examples

1) (123)=(13)(12)

2)(2345)=(25)(24)(23)

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