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Phone: (407) 823-6284;   Fax: (407) 823-6253;   MAP  207 2001-2002 MATHEMATICS COLLOQUIUM SERIES 03/025/02 Colloquium DR. PETER HILTON University of Central Florida ESSENTIAL EXTENSIONS Abstract:   In Honor of Niels Henrik Abel (1802-1829). We will discuss a very remarkable and rather elegant result in the theory of Abelian groups. Let B be an Abelian group and A a subgroup of B. We say that B is an essential extension of A if, for all non-zero subgroups H of B, we have non-empty intersection of H and A. Now every Abelian group A may be embedded in a divisible Abelian group A (i.e., an Abelian group in which elements admit division by arbitrary positive integers). We will justify the surprising statement:     The maximal essential extension of A is the minimal divisible extension of A.