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Bijection and mapping - ILL 1990 MON3

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September 18, 2010
functionalgebra unsolvedalgebra

Problem Statement

Let M={1,2,…,n}M = \{1, 2, \ldots, n\} and ϕ:M→M\phi : M \to M be a bijection.
(i) Prove that there exist bijections ϕ1,ϕ2:M→M\phi_1, \phi_2 : M \to M such that ϕ1⋅ϕ2=ϕ,ϕ12=ϕ22=E\phi_1 \cdot \phi_2 = \phi , \phi_1^2 =\phi_2^2=E, where EE is the identity mapping.
(ii) Prove that the conclusion in (i) is also true if MM is the set of all positive integers.
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