A Note on Hobby’s Theorem of Finite Groups

It is well known that the Frattini subgroups of any finite groups are nilpotent. If a finite group is not nilpotent, it is not the Frattini subgroup of a finite group. In this paper, we mainly discuss what kind of finite nilpotent groups cannot be the Frattini subgroup of some finite groups and give some results. Moreover, we generalize Hobby’s Theorem. 1. Introduction As we know, the Frattini subgroup of a finite group plays an important role in investigating the structure of finite groups. Many authors did this work, for example, the remarkable result of Burnside: let be a finite -group and let be a -invariant subgroup contained in the Frattini subgroup of . If is cyclic, then is also cyclic. In particular, if is a finite 2-group, then the Frattini subgroup cannot be a nonabelian group of order 8. Recently, in [1], Bo？ikov studied the next possible case, where is a finite 2-group and is nonabelian of order 16. He showed that in that case , where or and classified all such groups (Theorem A). On the other hand, we also know that the Frattini subgroup of a finite group is nilpotent. If a finite group is not nilpotent, it is not the Frattini subgroup of a finite group. In this paper, we go into what kind of finite nilpotent groups cannot be the Frattini subgroup of some finite groups and give some results in terms of seminormal subgroups of a finite group. Moreover, we generalize Hobby’s Theorem: a nonabelian -group with cyclic center cannot be the Frattini subgroup of any -group. Throughout the all groups mentioned are assumed to be finite groups. The terminology and notations employed agree with standard usage, as in [2] or [3]. We denote to indicate that is a normal subgroup of group . denotes that is a characteristic subgroup of group . denotes the Frattini subgroup of group . denotes the set of all primes dividing the order of group . 2. Basic Definitions and Preliminary Results In this section, we give one definition and some results that are needed in this paper. Definition 1 (see [4, Definition 1]). A subgroup of is seminormal in if there exists a subgroup such that and such that for every proper subgroup of , the product is a proper subgroup of . Lemma 2 (see [4, Proposition 5]). Let be a finite group. If -subgroup of is seminormal in , then permutes with every -subgroup of , where , are primes dividing the order of , . Lemma 3 (see [5, Lemma ]). Let be a -group. If there exists a normal subgroup such that , then . Lemma 4 (see [5, Lemma ]). Let be a -group and let be a normal subgroup of of order with . If and , then . Lemma 5. Let be a finite