Ein allgemeiner Sylowsatz in endlichen auflösbaren Gruppen.
A group has the endomorphism kernel property (EKP) if every congruence relation on is the kernel of an endomorphism on . In this note we show that all finite abelian groups have EKP and we show infinite series of finite non-abelian groups which have EKP.
We present a survey of results on word equations in simple groups, as well as their analogues and generalizations, which were obtained over the past decade using various methods: group-theoretic and coming from algebraic and arithmetic geometry, number theory, dynamical systems and computer algebra. Our focus is on interrelations of these machineries which led to numerous spectacular achievements, including solutions of several long-standing problems.