The Mathematics of Symmetry: Group Theory

Abstract

This chapter forms the start of an introduction to group theory, the mathematical language of symmetry. In it are discussed the most fundamental of the fundamental concepts and ideas of group theory. Here is a list of the more important concepts discussed in each section. Sect. 8.1: element, group, order, composition, closure, associativity, identity element, inverse element, noncommuting elements, commuting elements, Abelian (commutative) group, group structure, group table, realization. Sect. 8.2: mapping, object, image, many-to-one mapping, one-to-one mapping, into mapping, onto mapping, inverse mapping. Sect. 8.3: isomorphism. Sect. 8.4: homomorphism, kernel. Sect. 8.5: subgroup, proper subgroup.