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Endomorphisms on finite groups

Source: RMO 2008, Grade 12, Problem 4

April 30, 2008
inequalitiesgroup theoryabstract algebravectorsuperior algebrasuperior algebra unsolved

Problem Statement

Let G \mathcal G be the set of all finite groups with at least two elements. a) Prove that if G∈G G\in \mathcal G, then the number of morphisms f:G→G f: G\to G is at most nnp \sqrt [p]{n^n}, where p p is the largest prime divisor of n n, and n n is the number of elements in G G. b) Find all the groups in G \mathcal G for which the inequality at point a) is an equality.
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