Question and Answers Forum

All Questions      Topic List

Others Questions

Previous in All Question      Next in All Question      

Previous in Others      Next in Others      

Question Number 192340 by Mastermind last updated on 15/May/23

Prove that the order of any permuta− tion θ is the least common multiple of the length of its disjoint cycles. hi

Provethattheorderofanypermuta−tionθistheleastcommonmultipleofthelengthofitsdisjointcycles.hi

Answered by aleks041103 last updated on 15/May/23

let θ=c_1 ...c_k , where c_1 ,...,c_k are disjoint cycles. Since they are disjoint, they commute. ⇒θ^n =c_1 ^n ...c_k ^n By def. ∣θ∣=n, where n is the smallest positive integer such that θ^n =id. Since c_1 ,...,c_k are disjoint, c_1 ^n ,...,c_k ^n are also disjoint. ⇒θ^n =c_1 ^n ...c_k ^n =id⇔c_j ^n =id for ∀j=1,...,k it is obvious that c_j ^n =id ⇔ l(c_j )∣n, ∀j ⇒θ^n =id ⇔ lcm(l(c_1 ),...,l(c_k ))∣n ⇒∣θ∣=lcm(l(c_1 ),...,l(c_2 ))

letθ=c1...ck,wherec1,...,ckaredisjointcycles.Sincetheyaredisjoint,theycommute.⇒θn=c1n...cknBydef.∣θ∣=n,wherenisthesmallestpositiveintegersuchthatθn=id.Sincec1,...,ckaredisjoint,c1n,...,cknarealsodisjoint.⇒θn=c1n...ckn=id⇔cjn=idfor∀j=1,...,kitisobviousthatcjn=id⇔l(cj)∣n,∀j⇒θn=id⇔lcm(l(c1),...,l(ck))∣n⇒∣θ∣=lcm(l(c1),...,l(c2))

Commented by Mastermind last updated on 18/May/23

Thank you so much

Thankyousomuch

Terms of Service

Privacy Policy

Contact: info@tinkutara.com