MathDB
Romanian District Olympiad 2019 - Grade 12 - Problem 1

Source: Romanian District Olympiad 2019 - Grade 12 - Problem 1

March 16, 2019
endomorphismabtract algebraGroupssuperior algebra

Problem Statement

Let nn be a positive integer and GG be a finite group of order n.n. A function f:G→Gf:G \to G has the (P)(P) property if f(xyz)=f(x)f(y)f(z) ∀ x,y,z∈G.f(xyz)=f(x)f(y)f(z)~\forall~x,y,z \in G. <spanclass=′latex−bold′>(a)</span><span class='latex-bold'>(a)</span> If nn is odd, prove that every function having the (P)(P) property is an endomorphism. <spanclass=′latex−bold′>(b)</span><span class='latex-bold'>(b)</span> If nn is even, is the conclusion from <spanclass=′latex−bold′>(a)</span><span class='latex-bold'>(a)</span> still true?
Back to ProblemsView on AoPS