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We investigate the properties of groups g such that g/zg 'APPROX. ='C IND.P' x 'C IND.P' where zg denotes the center of g and 'C IND.P' the ciclic group of prime order p. These groups were suggested by the study of alternative loop rings and are also of interest to the isomorphism problem in group rings, since every complex representation is either of degree 1 or p. We also study the rational group algebras of these groups in connection with the isomorphism problem and apply these results to solve the corresponding problem for rational alternative loop algebras. Finally we consider the problem of finding subgroups of units of finite index in the units of the integral group rings of such groups and determine generators of these subgroups, or the whole group of units, in some special cases