A non-simplicity criterion of finite groups

Characterization of groups is normally done in two ways: by considering the structure of the group directly or by looking at the “local” properties of the group. Group theory and character theory are necessary for both methods. This research is a continuation of the characterization problem by looking at a local property of an abstract finite group G. We consider the following problem: Let G be a finite group with H a maximal subgroup satisfying (i) H = XP, where x = R x S, R has odd order, S is a 2-group, P = is cyclic of order 3, And t2 = (xt)2 = 1. (ii) X acts fixed-point-free on S; S # 1. (iii) H is the only maximal subgroup of G containing K = (R x S) P and [W1z(s)]>4. It is established that such a finite group G is non-simple. In one of the recent works, A.A. Liggonah considers a finite group G with a maximal subgroup H satisfying the hypotheses given in [24] and establishes that G is non-simple. In particular, he considers P to be an elementary abelian sylow 3-subgroup of order 9. G. Higman establishes, for G satisfying the same conditions as in [24] but with P being a cyclic sylow 3-subgroup of order 3, that G is non-simple; (refer [16].