Monoids and groups
This chapter is devoted to algebraic systems of one binary operation. Groups have been used many times before in the text, as permutations, as modules (a commutative group with an exterior ring multiplication), and elsewhere, but here we organize the systems of one binary operation into one chapter and give some additional specialized results. The most general algebraic system we consider in this chapter is the monoid, a set with one associative binary operation. Some special monoids discussed are unitary monoids, cancellative monoids, and groups. We discuss subsystems, quotient systems, morphisms, and the fundamental morphism theorem. In Sections 9.4 and 9.5 we study more specialized results available for groups alone. We study cyclic groups and connect the order of an element with this concept. Several topics such as center, normalizer, coniugacy classes are all organized around inner automorphisms (φ a (x) = a − xa). We then apply these results to answer some questions about elements of prime order. We offer some standard theorems relating direct products and Cartesian products of groups. We define simple groups and solvable groups. We give an inductive proof of the fundamental theory of Abelian groups which is subsumed in Chapter 10 by the direct sum resolution of a finitely generated module.
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