Finite groups with formation subnormal primary subgroups of bounded exponent

Abstract

Let 𝔘k be the class of all supersoluble groups in which exponents are not divided by the (k+1)-th powers of primes. We investigate the classes w𝔘k and v𝔘k that contain all finite groups in which every Sylow and, respectively, every cyclic primary subgroup is 𝔘k-subnormal. We prove that w𝔘k and v𝔘k are subgroup-closed saturated formations and obtain the characterizations of these formations.