Suppose α and β are permutations ofS=a1,a2,...,am,b1,b2,...,bn,c1,c2,...,ckwhere the c’s are left fixed by both α and β
[To show αβ(x)=βα(x)∀ x ∈ S]
ifx=ai for some i,since βfixes all a elements,
(αβ)(ai)=α(β(ai))=α(ai)=ai+1(with am+1=a1)and
(βα)(ai)=β(α(ai))=β(ai)=ai+1,so αβ=βα on the a elements.
A similar argument shows αβ=βα on the b elements.
Since α and β both fix the c elements,
(αβ)(ci)=α(β(ci))=α(ci)=ci and
(βα)(ci)=β(α(ci))=β(ci)=ci.
thus αβ(x)=βα(x)∀ x ∈ S
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