Investigation of Some Subgroups in Determining the Frattini Subgroup of Non-Abelian Finite Groups

Frattini subgroup, Φ(G), of a group G is the intersection of all the maximal subgroups of G, or else G itself if G has no maximal subgroups. If G is a p-group, then Φ(G) is the smallest normal subgroup N such the quotient group G/N is an elementary abelian group. It is against this background that the concept of p-subgroup and fitting subgroup play a significant role in determining Frattini subgroup (especially its order) of dihedral groups. A lot of scholars have written on Frattini subgroup, but no substantial relationship has so far been identified between the parent group G and its Frattini subgroup Φ(G) which this tries to establish using the approach of Jelten B. Napthali who determined some internal properties of non abelian groups where the centre Z(G) takes its maximum size.

A generalized Frattini subgroup

In the work of Beidleman and Smith [On Frattini-like subgroups, Glasgow Math. J. 35 (1993) 95–98], the following question was raised: “If [Formula: see text] is a subnormal subgroup of a finite group [Formula: see text] containing [Formula: see text], then whether the supersolvability of [Formula: see text] follows the supersolvability of [Formula: see text]”. This problem was considered in works of Selkin [Maximal Subgroups in the Theory of Classes of Finite Groups (Belaruskaya, Navuka, 1997)], Skiba [On the intersection of all maximal [Formula: see text]-subgroups of a finite group, Prob. Phys. Math. Tech. 3(4) (2010) 56–62], Ballester-Bolinches [On [Formula: see text]-subnormal subgroups and Frattini-like subgroups of a finite group, Glasgow Math. J. 36 (1994) 241–247] and many other authors (see monograph [Maximal Subgroups in the Theory of Classes of Finite Groups (Belaruskaya, Navuka, 1997)]). In this paper, we give the answer to the more general question: “Let [Formula: see text] be a local formation. If [Formula: see text] is a subnormal subgroup of a group [Formula: see text], then in what case [Formula: see text] will follow from [Formula: see text]”.

On Finite Quasi-Core-p p-Groups

Given a positive integer n, a finite group G is called quasi-core-n if ⟨ x ⟩ / ⟨ x ⟩ G has order at most n for any element x in G, where ⟨ x ⟩ G is the normal core of ⟨ x ⟩ in G. In this paper, we investigate the structure of finite quasi-core-p p-groups. We prove that if the nilpotency class of a quasi-core-p p-group is p + m , then the exponent of its commutator subgroup cannot exceed p m + 1 , where p is an odd prime and m is non-negative. If p = 3 , we prove that every quasi-core-3 3-group has nilpotency class at most 5 and its commutator subgroup is of exponent at most 9. We also show that the Frattini subgroup of a quasi-core-2 2-group is abelian.

On metanilpotent groups satisfying the minimal condition on normal subgroups

A comprehensive account is given of the theory of metanilpotent groups with the minimal condition on normal subgroups. After reviewing classical material, many new results are established relating to the Fitting subgroup, the Hirsch–Plotkin radical, the Frattini subgroup, splitting and conjugacy, the Schur multiplier, Sylow structure and the maximal subgroups. Module theoretic and homological methods are used throughout.

A note on an “Anzahl” theorem of P. Hall

Let [Formula: see text] be a prime and [Formula: see text] a finite [Formula: see text]-group. Hall proved [Formula: see text] for [Formula: see text], where [Formula: see text] is the number of subgroups of index [Formula: see text] which do not contain the Frattini subgroup of [Formula: see text]. We determine the minimal values of [Formula: see text] for [Formula: see text] and [Formula: see text].

On noninner automorphisms of finite p-groups that fix the Frattini subgroup elementwise

Let [Formula: see text] be a finite [Formula: see text]-group and let [Formula: see text]. In this paper we show that if [Formula: see text] lies in the second center [Formula: see text] of [Formula: see text], then [Formula: see text] admits a noninner automorphism of order [Formula: see text], when [Formula: see text] is an odd prime, and order [Formula: see text] or [Formula: see text], when [Formula: see text]. Moreover, the automorphism can be chosen so that it induces the identity on the Frattini subgroup [Formula: see text]. When [Formula: see text], this reduces the verification of the well-known conjecture that states every finite nonabelian [Formula: see text]-group [Formula: see text] admits a noninner automorphism of order [Formula: see text] to the case in which [Formula: see text] where [Formula: see text]. In addition, it follows that if [Formula: see text] is a finite nonabelian [Formula: see text]-group, [Formula: see text], such that [Formula: see text] is a cohomologically trivial [Formula: see text]-module, then [Formula: see text] satisfies the above mentioned condition, and as a consequence we show that the order of [Formula: see text] is at least [Formula: see text].