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