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About 25 years ago, Persi Diaconis asked the following question: Is it possible to develop a "K-theory of finite groups using finite generating sequences" in a fashion analogous to what is done in the classical situation? New properties and questions about finite groups that arise from studying various approaches to answering this question will be developed. Natural connections with older work of Tarski, the Neumanns, and others will appear. Many unsolved problems about finite groups, both old and new, appear in a natural way. Diaconis' question seems to have generated many interesting problems and gives an interpretation to how some of the current work in finite group theory might fit into a larger picture. Prerequisites: Basic group theory, e.g., MATH 4340 or MATH 6310.