Continuing their exposition of algebra begun in Volume 1, in this book the authors present the complexities associated with the study of groups. The main topics studied are: integral and algebraic elements, rings of matrices, quaternions with applications to SU (2,C) and SO (3,R) as well as the rings of endomorphisms and fractions and ideal theory. As with its predecessor, this volume is filled with supportive material including purposeful examples and instructive exercises with a focus on the connections with number theory.