Question #141709
Any two disjoint permutation commute
1
Expert's answer
2020-11-02T20:41:05-0500

Any permutation is regarded as an element of the symmetric group S(X)S(X) where XX is a set of elements on which we permute.Hence let σ,τ\sigma,\tau be two such permutations. They are disjoint means, the elements not fixed by σ\sigma are fixed by τ\tau and vise-versa. We need to show στ=τσ.\sigma\tau=\tau\sigma. Let aa be an element of the set XX. Now if σ,τ\sigma,\tau both fixes a,a, then στ(a)=σ(a)=a=τσ(a)\sigma\tau(a)=\sigma(a)=a=\tau\sigma(a) . Now let τ\tau doesn't fix it . Then by above discussion σ\sigma has to fix it. Let τ(a)=b\tau(a)=b . Then τσ(a)=τ(a)=b.\tau\sigma(a)=\tau(a)=b. Again, στ(a)=σ(b).\sigma\tau(a)=\sigma(b). Now τ\tau is a permutation i.e. a bijection. Hence τ(b)τ(a)=b.\tau(b)\neq\tau(a)=b. Hence τ\tau doesn't fix b.b. Hence σ\sigma must fix bb or σ(b)=b.\sigma(b)=b. Hence στ(a)=τσ(a).\sigma\tau(a)=\tau\sigma(a). Similarly the reult holds for the case when σ\sigma doesn't fix a.a. Hence τσ=στ.\tau\sigma=\sigma\tau. Hence done


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