Can you please show me the proof of this theorem?

" Every permutation under Sn where n>1 is a product of two cycles"

thanks (Giggle)

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- Sep 3rd 2010, 12:17 AMjherv05Permutation Groups
Can you please show me the proof of this theorem?

" Every permutation under Sn where n>1 is a product of two cycles"

thanks (Giggle)

- Sep 3rd 2010, 04:04 PMRoam
Clearly the identity can be expressed as $\displaystyle (12)(12)$ and it is a product of 2-cycles. And since you know that every permutation of a finite set can be written as a cycle or as a product of disjoint cycles (proof?), every permutation can be expressed in the form:

$\displaystyle (a_1 a_2...a_k)(b_1b_2...b_t)...(c_1c_2...c_s)$

then you will have:

$\displaystyle

(a_1a_k)(a_1a_{k-1})...(a_1a_2)(b_1b_t)(b_1b_{t-1})...(b_1b_2)...(c_1c_s)(c_1c_{s-1})...(c_1c_2)

$