# Permutation Groups

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• September 3rd 2010, 01:17 AM
jherv05
Permutation 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)
• September 3rd 2010, 05:04 PM
Roam
Clearly the identity can be expressed as $(12)(12)$ 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:

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

then you will have:

$
(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)
$