Around 1980, G. Mason announced the classification of a certain subclass of an important class of finite simple groups known as 'quasithin groups'. The classification of the finite simple groups depends upon a proof that there are no unexpected groups in this subclass. Unfortunately Mason neither completed nor published his work. In the ""Main Theorem"" of this two-part book (Volumes 111 and 112 in the AMS series, ""Mathematical Surveys and Monographs"") the authors provide a proof of a stronger theorem classifying a larger class of groups, which is independent of Mason's arguments.In particular, this allows the authors to close this last remaining gap in the proof of the classification of all finite simple groups. An important corollary of the Main Theorem provides a bridge to the program of Gorenstein, Lyons, and Solomon (Volume 40 in the AMS series, ""Mathematical Surveys and Monographs"") which seeks to give a new, simplified proof of the classification of the finite simple groups. Part I (the current volume) contains results which are used in the proof of the Main Theorem. Some of the results are known and fairly general, but their proofs are scattered throughout the literature; others are more specialized and are proved here for the first time. Part II of the work (Volume 112) contains the proof of the Main Theorem, and the proof of the corollary classifying quasithin groups of even type. The book is suitable for graduate students and researchers interested in the theory of finite groups.