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surjective homomorphisms

Source: miklos schweitzer 2011 q4

August 29, 2021
abstract algebraSurjectivegroup theory

Problem Statement

Let G, H be two finite groups, and let φ,ψ:G→H\varphi, \psi: G \to H be two surjective but non-injective homomorphisms. Prove that there exists an element of G that is not the identity element of G but whose images under φ\varphi and ψ\psi are the same.
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