This is one of a series of publications written to supplement the secondary school School Mathematics Study Group program. This booklet will be most useful for enrichment at the eleventh and twelfth grade levels. It treats algebraic structures as abstract mathematical systems and introduces such important ideas as group, non-abelian group, field, and subfield. Proofs are rigorous, but not tedious. Answers to the problems are found in the back of the book. As background, the reader needs to be familiar with the following sets of numbers: integers, rationals, reals, and complex numbers. (Author/RH)