Some properties of the supersoluble formation and the supersoluble residual of a group

Purpose: In this paper, We determine the finite group G = HK such that K is a supersoluble subgroup of G, and H is not a supersoluble subgroup of G. Methods: Let p, q, r be primes such that p q r, and p, q are not a divisor of r − 1, and p is not a divisor of q − 1. Let X be a group of order p, and let F = GF(q) and L = GF(r) such that the filed F contains a primitive pth root of unity. Let V be a simple FX-module, and let Y = V ɣ X and W also be a faithful simple LY-module. Let G = W ɣ Y, H = W ɣ X, and K = W ɣ V. Results: Then, we determine that K is a supersoluble subgroup of G, and H is not a supersoluble subgroup of G. Conclusions: We characterize the supersoluble residual of group G.