This is the first of three volumes of a comprehensive and elementary treatment of finite -group theory. Topics covered in this monograph include: (a) counting of subgroups, with almost all main counting theorems being proved, (b) regular -groups and regularity criteria, (c) -groups of maximal class and their numerous characterizations, (d) characters of -groups, (e) -groups with large Schur multiplier and commutator subgroups, (f) ( ‒1)-admissible Hall chains in normal subgroups, (g) powerful -groups, (h) automorphisms of -groups, (i) -groups all of whose nonnormal subgroups are cyclic, (j) Alperin's problem on abelian subgroups of small index.