On necessary and sufficient conditions of simply reducibility of wreath product of finite groups

A finite group is considered to be real if all the values of its complex irreducible characters lie in the field of real numbers. We note that the above reality condition is equivalent to the fact that each element of the group is conjugate to its inverse. A finite group is called simply reducible or a SR-group if it is real and all the coefficients of the decomposition of the tensor product of any two of its irreducible characters are zero or one. The notion of a SR-group arose in the paper of R. Wiener in connection with the solution of eigenvalue problems in quantum theory. At present, there is a sufficient amount of literature on the theory of SR-groups and their applications in physics. The simplest examples of SR-groups are elementary Abelian 2-groups, dihedral groups, and generalized quaternion groups. From the point of view of a group theory questions of interest are connected first of all with the structure of simply reducible groups. For example A. I. Kostrikin formulated the following question: how to express the belonging of a finite group to the class of SR-groups in terms of the structural properties of the group itself. Also, for a long time it was not known whether a simply reducible group is solvable (S. P. Stunkov's question). A positive answer to the last question was obtained in the works of L. S. Kazarin, V. V. Yanishevskiy, and E. I. Chankov. Questions concerning the portability of the properties of a group to subgroups, factor groups, and also their preservation in the transition to direct (Cartesian) and semidirect products or wreath products are always of interest. The paper proves that the reality of H is the necessary condition of simply reducibility of the wreath product of the finite group H with the finite group K and the group K must be an elementary Abelian 2-group. We also indicate sufficient conditions for simply reducibility of a wreath product of a simply reducible group with a cyclic group of order