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Number of the members of the set is divisible by p [Iran 92]

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November 28, 2010
functioninductionnumber theory proposednumber theory

Problem Statement

Let X≠∅X \neq \varnothing be a finite set and let f:X→Xf: X \to X be a function such that for every x∈Xx \in X and a fixed prime pp we have fp(x)=x.f^p(x)=x. Let Y={x∈X∣f(x)≠x}.Y=\{x \in X | f(x) \neq x\}. Prove that the number of the members of the set YY is divisible by p.p.
Note. fp(x)=x=f(f(f(⋯((f⏟p times(x))⋯ ))).{f^p(x)=x = \underbrace{f(f(f(\cdots ((f}_{ p \text{ times}}(x) ) \cdots )))} .
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