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This text provides an easy to follow introduction to the theory of groups and of group characters. The author presents group theory through the Sylow Theorems and uses many worked-out examples, including the full subgroup structure of A5. Representations and characters are worked out with numerous character tables plus real and induced characters that lead to the table for S5. The book includes sections that provide the mathematical basis for some of the applications of group theory in spectroscopy and molecular structure. It is also full of exercises that five the reader a sense of the applications to larger projects.