Finite Groups with Some 𝑆𝐸-Supplemented Subgroups

Abstract

Let 𝐻 be a subgroup of a finite group 𝐺, 𝑝 a prime dividing the order of 𝐺, and 𝑃 a Sylow 𝑝-subgroup of 𝐺 for prime 𝑝. We say that 𝐻 is 𝑆𝐸-supplemented in 𝐺 if there is a subgroup 𝐾 of 𝐺 such that 𝐺=𝐻𝐾 and 𝐻∩𝐾≤𝐻_{𝑠𝑒𝐺,} where 𝐻_𝑠𝑒𝐺 denotes the subgroup of 𝐻 generated by all those subgroups of 𝐻 which are 𝑆-quasinormally embedded in 𝐺. In this paper, we characterize 𝑝-nilpotency and supersolvability of 𝐺 under the assumption that all maximal subgroups of 𝑃 are 𝑆𝐸-supplemented in 𝑁_𝐺(𝑃).

1. Introduction

Throughout only finite groups are considered. Let 𝜋(𝐺) stand for the set of all prime divisors of the order of a group 𝐺. 𝒰 denotes the class of all supersolvable groups. 𝐻 char 𝐺 means 𝐻 is a characteristic subgroup of 𝐺. We use conventional notions and notation, as in Robinson [1].

Let ℱ be a class of groups. ℱ is called a formation provided that (1) if 𝐺∈ℱ and 𝐻⊲𝐺, then 𝐺/𝐻∈ℱ and (2) if 𝐺/𝑀 and 𝐺/𝑁 are in ℱ, then 𝐺/𝑀∩𝑁 is in ℱ for all normal subgroups 𝑀, 𝑁 of 𝐺. A formation ℱ is said to be saturated if 𝐺/Φ(𝐺)∈ℱ implies that 𝐺∈ℱ.

Two subgroups 𝐻 and 𝐾 of a group 𝐺 are said to be permutable if 𝐻𝐾=𝐾𝐻. 𝐻 is said to be 𝑆-quasinormal in 𝐺 if 𝐻 permutes with every Sylow subgroup of 𝐺, that is, 𝐻𝑃=𝑃𝐻 for any Sylow subgroup 𝑃 of 𝐺. This concept was introduced by Kegel in [2] and has been studied widely by many authors, such as [3, 4]. An interesting question in the theory of finite groups is to determine the influence of the embedding properties of members of some distinguished families of subgroups of a group on the structure of the group. Recently, Ballester-Bolinches and Pedraza-Aguilera [5] generalized 𝑆-quasinormal subgroups to 𝑆-quasinormally embedded subgroups. 𝐻 is said to be 𝑆-quasinormally embedded in 𝐺 provided every Sylow subgroup of 𝐻 is a Sylow subgroup of some 𝑆-quasinormal subgroup of 𝐺. By applying this concept, Ballester-Bolinches and Pedraza-Aguilera got new criteria for supersolvability of groups.

A subgroup 𝐻 of a group 𝐺 is called to be complemented in 𝐺 if 𝐺 has a subgroup 𝐾 such that 𝐺=𝐻𝐾 and 𝐻∩𝐾=1. 𝐻 is called to be supplemented in 𝐺 if there exists a subgroup 𝑇 of 𝐺 such that 𝐺=𝐻𝑇. Obviously, a complemented subgroup is a special supplemented subgroup. In recent years, it has been of interest to use supplementation properties of subgroups to characterize properties of a group. For example, the definition of a weakly 𝑆-supplemented subgroup introduced by Skiba in [6]. 𝐻 is said to be weakly 𝑆𝐸-supplemented in 𝐺 if 𝐺 has a subgroup 𝑈 such that 𝐻𝑈=𝐺 and 𝐻∩𝑈≤𝐻_𝑠𝐺, where 𝐻_𝑠𝐺 is the largest 𝑆-quasinormal subgroup of 𝐺 contained in 𝐻. Using this concept, many meaningful results have been obtained, such as [7–9]. More recently, the concept of 𝑆𝐸-supplemented subgroups was introduced as follows.

Definition 1 (see [10, Definition 1.2]). Let 𝐻 be a subgroup of 𝐺. We say that 𝐻 is 𝑆𝐸-supplemented in 𝐺 if there exists some subgroup 𝑇 of 𝐺 such that 𝐻𝑇=𝐺 and 𝐻∩𝑇≤𝐻_𝑠𝑒𝐺, where 𝐻_𝑠𝑒𝐺 denotes the subgroup generated by all the subgroups of 𝐻 which are 𝑆-quasinormally embedded in 𝐺. We call 𝐻_𝑠𝑒𝐺 the 𝑆𝐸-core of 𝐻 in 𝐺.

Clearly, normal subgroups, 𝑆-quasinormal subgroups, 𝑆-quasinormally embedded subgroups, and weakly 𝑆-supplemented subgroups are all 𝑆𝐸-supplemented subgroups. But the converse does not hold in general (see [11]). Based on the observation of the above concepts, we note that supplementation of some families of subgroups of a group has a strong influence on its structure.

On the other hand, the normalizer of a Sylow subgroup of a group takes an important role in studying the structure of a group. Let 𝑃 be a Sylow subgroup of a group 𝐺; it is natural to consider the structure of 𝐺 if some properties of the normalizer 𝑁_𝐺(𝑃) of 𝑃 are known. A classical result in this orientation is attributed to Burnside's Theorem [1, Theorem  10.1.8]. Later, Hall [12] extended it as follows: if each 𝑝^′-element of 𝑁_𝐺(𝑃) does commute with all elements of 𝑃 and if also the class size of 𝑃 is less than 𝑝, then 𝐺 is 𝑝-nilpotent. In short, it is of significance to research into the structure of finite groups from properties of the normalizer of a Sylow subgroup of a group. In this paper, under the assumption that all maximal subgroups of a Sylow subgroup 𝑃 are 𝑆𝐸-supplemented in 𝑁_𝐺(𝑃), we give some new characterizations on the structure of 𝐺.

2. Preliminaries

For the sake of convenience, we first cite some known results in the literature which will be useful in the following.

Lemma 2 (see [13, Lemma 2.1]). Let 𝐻 and 𝐾 be subgroups of a group 𝐺.(1)If 𝐻 is 𝑆-quasinormal in 𝐺 and 𝐻≤𝑀≤𝐺, then 𝐻 is 𝑆-quasinormal in 𝑀.(2)Let 𝑁⊲𝐺 and 𝐻 be 𝑆-quasinormal in 𝐺. Then 𝐻𝑁 is 𝑆-quasinormal in 𝐺 and 𝐻𝑁/𝑁 is 𝑆-quasinormal in 𝐺/𝑁.(3)If 𝐻 is 𝑆-quasinormal in 𝐺, then 𝐻 is subnormal in 𝐺.(4)If both 𝐻 and 𝐾 are 𝑆-quasinormal subgroups of 𝐺, then both 𝐻∩𝐾 and ⟨𝐻,𝐾⟩ are 𝑆-quasinormal subgroups of 𝐺.

Lemma 3 (see [10, Lemma 2.6]). Suppose that 𝑁 is a normal subgroup of 𝐺 and 𝐻≤𝐾≤𝐺. Then(1)𝐻_𝑠𝑒𝐺⊴𝐻;(2)𝐻_𝑠𝑒𝐺≤𝐻_𝑠𝑒𝐾; (3)𝐻_𝑠𝑒𝐺𝑁/𝑁≤(𝐻𝑁/𝑁)_{𝑠𝑒(𝐺/𝑁)};(4)If (|𝑁|,|𝐻|)=1, then 𝐻_𝑠𝑒𝐺𝑁/𝑁=(𝐻𝑁/𝑁)_{𝑠𝑒(𝐺/𝑁)}.

Lemma 4 (see [10, Lemma 2.7]). Let 𝐻 be a subgroup of a group 𝐺.(1)If 𝐻 is 𝑆𝐸-supplemented in 𝐺 and 𝐻≤𝑀≤𝐺, then 𝐻 is 𝑆𝐸-supplemented in 𝑀.(2)Let 𝑁⊲𝐺 and 𝑁≤𝐻. If 𝐻 is 𝑆𝐸-supplemented in 𝐺, then 𝐻/𝑁 is 𝑆𝐸-supplemented in 𝐺/𝑁.(3)Let 𝜋 be a set of primes, 𝐻 a 𝜋-subgroup of 𝐺, and 𝑁 a normal 𝜋^′-subgroup of 𝐺. If 𝐻 is 𝑆𝐸-supplemented in 𝐺, then 𝐻𝑁/𝑁 is 𝑆𝐸-supplemented in 𝐺/𝑁.

Lemma 5 (see [14]). Let 𝐺 be a group.(1)If 𝑃 is an 𝑆-quasinormal 𝑝-subgroup of 𝐺 for some prime 𝑝, then 𝑁_𝐺(𝑃)≥𝑂^𝑝(𝐺).(2)Suppose that 𝐻 is a 𝑝-subgroup of 𝐺 contained in 𝑂_𝑝(𝐺). If 𝐻 is 𝑆-quasinormally embedded in 𝐺, then 𝐻 is 𝑆-quasinormal in 𝐺.

Lemma 6. Let 𝐻 be a subgroup of a group 𝐺. If 𝐻 is 𝑆𝐸-supplemented in 𝐺 and 𝐻≤𝑂_𝑝(𝐺), then 𝐻 is weakly 𝑆𝐸-supplemented in 𝐺.

Proof. In fact, 𝐻_𝑠𝑒𝐺=⟨𝐻₁,𝐻₂,…,𝐻_𝑠⟩, where 𝐻_𝑖 (𝑖=1,2,…,𝑠) is 𝑆-quasinormally embedded in 𝐺. Since 𝐻≤𝑂_𝑝(𝐺), we have that 𝐻_𝑖 is 𝑆-quasinormal in 𝐺 by Lemma 5(2). Thus, 𝐻_𝑠𝑒𝐺≤𝐻_𝑠𝐺, where 𝐻_𝑠𝐺 is the largest 𝑆-quasinormal subgroup of 𝐺 contained in 𝐻. Consequently, 𝐻 is weakly 𝑆𝐸-supplemented in 𝐺.

Lemma 7 (see [15, Lemma 2.1]). Let 𝐺 be a group. If 𝐴 is subnormal in 𝐺 and 𝐴 is a 𝜋-subgroup of 𝐺, then 𝐴≤𝑂_𝜋(𝐺).

From Lemma 6, the following Lemma is a direct corollary of Lemma 3.1 in [7].

Lemma 8. Suppose 𝐺=𝑃𝑄, where 𝑃 is a normal Sylow 𝑝-subgroup and 𝑄 a Sylow 𝑞-subgroup of 𝐺. If (|𝐺|,𝑝−1)=1 and if also every maximal subgroup of 𝑃 is 𝑆𝐸-supplemented in 𝐺, then 𝐺 is 𝑝-nilpotent.

3. Main Results

Theorem 9. Let 𝐺 be a group and assume 𝑝 is a prime dividing the order of 𝐺 with (|𝐺|,𝑝−1)=1. If there exists a Sylow 𝑝-subgroup 𝑃 of 𝐺 such that every maximal subgroup of 𝑃 is 𝑆𝐸-supplemented in 𝑁_𝐺(𝑃) and if also 𝑃^′ is 𝑆-quasinormal in 𝐺, then 𝐺 is 𝑝-nilpotent.

Proof. Suppose that the result is not true and let 𝐺 be a counterexample of minimal order. We will derive a contradiction in several steps.
(1)  𝑂_𝑝(𝐺)≥𝑃^′≠1. Let 𝑄∈Syl_𝑞(𝑁_𝐺(𝑃)), where 𝑞 is a prime number dividing |𝑁_𝐺(𝑃)| and different from 𝑝. We can see that all maximal subgroups of 𝑃 are 𝑆𝐸-supplemented in 𝑃𝑄 by Lemma 4. Then 𝑃𝑄 meets the hypothesis of Lemma 8. It follows that 𝑃𝑄 is 𝑝-nilpotent and thus 𝑄≤𝐶_𝐺(𝑃). We know that all 𝑝^′-elements of 𝑁_𝐺(𝑃) are contained in 𝐶_𝐺(𝑃). If 𝑃 is abelian, then 𝑁_𝐺(𝑃)=𝐶_𝐺(𝑃), which yields that 𝐺 is 𝑝-nilpotent from Burnside's theorem [1, Theorem  10.1.8], which is a contradiction. Thus we may assume that 𝑃^′≠1. Since 𝑃^′ is 𝑆-quasinormal in 𝐺, thus 𝑃^′ is subnormal in 𝐺 by Lemma 2. It follows from Lemma 7 that 1≠𝑃^′≤𝑂_𝑝(𝐺).
(2)  For any Normal Subgroup 𝑁 of 𝐺 Contained in 𝑃, 𝐺/𝑁 is 𝑝-Nilpotent and 𝐺 is Solvable. It is clear that (|𝐺/𝑁|,𝑝−1)=1. For any maximal subgroup 𝑃₁/𝑁 of 𝑃/𝑁, 𝑃₁ is a maximal subgroup of 𝑃. From this hypothesis, 𝑃₁ is 𝑆𝐸-supplemented in 𝑁_𝐺(𝑃) and 𝑃^′ is 𝑆-quasinormal in 𝐺. It follows by Lemma 4 that 𝑃₁/𝑁 is 𝑆𝐸-supplemented in 𝑁_𝐺(𝑃)/𝑁=𝑁_𝐺/𝑁(𝑃/𝑁), and (𝑃/𝑁)^′=𝑃^′𝑁/𝑁 is 𝑆-quasinormal in 𝐺/𝑁 by Lemma 2. As a result, 𝐺/𝑁 meets the hypothesis of the theorem. The choice of 𝐺 yields that 𝐺/𝑁 is 𝑝-nilpotent. Let 𝑀/𝑁 be a normal 𝑝-complement of 𝐺/𝑁. If 𝑝=2, then 𝑀/𝑁 is a group of odd order. It follows from the Feit-Thompson theorem which asserts that every group of odd order is solvable, so is 𝑀. We note that 𝐺/𝑀 is a 2-group, and so 𝐺 is solvable. If 𝑝≠2, then 𝐺 is a group of odd order by (|𝐺|,𝑝−1)=1. Similarly, it deduces that 𝐺 is solvable, too.
(3) |𝐺|=𝑝^𝑎𝑞^𝑏 for Some Prime 𝑞≠𝑝. Since 𝐺 is solvable, there exists a Sylow system 𝑃₁={𝑃,𝑃₂,…,𝑃_𝑠} of 𝐺 with 𝐺_𝑖=𝑃𝑃_𝑖 for 2≤𝑖≤𝑠. By Lemmas 2 and 4, the hypothesis still holds for every 𝐺_𝑖. If |𝜋(𝐺)|>2, then 𝐺_𝑖<𝐺 and thus 𝐺_𝑖 is 𝑝-nilpotent by the choice of 𝐺, whence 𝑃_𝑖⊴𝐺_𝑖, thereby meaning that 𝑃 normalizes 𝑃_𝑖 for each 2≤𝑖≤𝑠. Therefore 𝐺 is 𝑝-nilpotent and 𝐾=𝑃₂𝑃₃⋯𝑃_𝑠 is a normal 𝑝-complement of 𝐺, which is a contradiction. Now, we may assume that |𝐺|=𝑝^𝑎𝑞^𝑏.
(4) The Final Contradiction. Let 𝑁 be a minimal normal subgroup of 𝐺. Because 𝑃^′ is 𝑆-quasinormal in 𝐺 and 𝑁⊴𝐺, 𝑃^′𝑁/𝑁 is 𝑆-quasinormal in 𝐺/𝑁 by Lemma 2. Now we consider the quotient group 𝐺/𝑁. If 𝑁 is a 𝑞-group, then 𝑃𝑁/𝑁∈Syl_𝑝(𝐺/𝑁), and for any maximal subgroup 𝑀/𝑁 of 𝑃𝑁/𝑁, we have 𝑀=𝑃₁𝑁, where 𝑃₁ is a maximal subgroup of 𝑃. From this hypothesis, 𝑃₁ is 𝑆𝐸-supplemented in 𝑁_𝐺(𝑃). Then there is a subgroup 𝑇 of 𝐺 such that 𝑁_𝐺(𝑃)=𝑃₁𝑇 and 𝑃₁∩𝑇≤(𝑃₁)_{𝑠𝑒𝑁𝐺(𝑃)}. Since 𝑃∈Syl_𝑝(𝐺), we know that 𝑁_𝐺/𝑁𝑃𝑁𝑁=𝑁_𝐺(𝑃)𝑁𝑁=𝑃₁𝑁𝑁𝑇𝑁𝑁. (1) As (|𝑃₁|,|𝑁|)=1, 𝑃₁=𝑃∩𝑇𝑁¹⋅|𝑇𝑁|_𝑝𝑃₁𝑇𝑁_𝑝=𝑃₁⋅|𝑇|_𝑝𝑁_𝐺(𝑃)𝑁_𝑝=𝑃₁⋅|𝑇|_𝑝𝑁_𝐺(𝑃)_𝑝=𝑃₁∩𝑇. (2) This means that 𝑃₁∩𝑇𝑁=𝑃₁∩𝑇, and thus we have 𝑃₁𝑁𝑁∩𝑇𝑁𝑁=𝑃₁𝑁∩𝑇𝑁𝑁=𝑃₁∩𝑇𝑁𝑁𝑁=𝑃₁∩𝑇𝑁𝑁≤𝑃_{1𝑠𝑒𝑁𝐺(𝑃)}𝑁𝑁. (3) By Lemma 3, it follows that (𝑃₁)_{𝑠𝑒𝑁𝐺(𝑃)}𝑁/𝑁≤(𝑃₁𝑁/𝑁)_{𝑠𝑒(𝑁𝐺/𝑁(𝑃𝑁/𝑁))} and thus 𝐺/𝑁 meets the hypothesis of our theorem. The choice of 𝐺 implies that 𝐺/𝑁 is 𝑝-nilpotent, and so is 𝐺, contradicting the fact that 𝐺 is a counterexample of minimal order. Consequently, we may assume that 𝑁 is a 𝑝-group and thus 𝑁≤𝑃. It follows by (2) that 𝐺/𝑁 is 𝑝-nilpotent. Hence we may assume that 𝑁 is the unique minimal normal subgroup of 𝐺 and 𝑁≰Φ(𝐺). These mean that Φ(𝐺)=1 and 𝑂_𝑝(𝐺)=𝐹(𝐺)=𝑁.
As 𝑃^′ is 𝑆-quasinormal in 𝐺, we know that 𝑁_𝐺(𝑃^′)≥𝑂^𝑝(𝐺) by Lemma 5. Since 𝑃 normalizes 𝑃^′, we get that 𝑃^′⊴𝐺, then 𝑃^′=𝑂_𝑝(𝐺)=𝑁 since 𝑁 is the unique minimal normal subgroup of 𝐺. It follows from (2) that 𝐺/𝑂_𝑝(𝐺) is 𝑝-nilpotent; thus 𝑂_𝑝(𝐺)𝑄⊴𝐺, where 𝑄∈Syl_𝑞(𝐺). Since 𝑂_𝑝(𝐺)𝑄∩𝑃=𝑂_𝑝(𝐺)=𝑃^′≤Φ(𝑃),  𝑂_𝑝(𝐺)𝑄 is 𝑝-nilpotent by Tate's Theorem [16, Theorem 4.4.7]. Thus 𝑄 char 𝑂_𝑝(𝐺)𝑄⊴𝐺 which yields that 𝑄⊴𝐺; that is, 𝐺 is 𝑝-nilpotent, a contradiction. The final contradiction completes the proof.

Remark 10. In Theorem 9, the condition that 𝑃^′ is 𝑆-quasinormal in 𝐺 cannot be removed.

Example 11. Let 𝐺=PSL₂(𝑞), where 𝑞>1 and 𝑞≡±1(mod8). Let 𝑃 be a Sylow 2-subgroup of 𝐺. By [16, II, Theorem 8.27], we know that 𝑃 is self-normalizing in PSL₂(𝑞). Clearly, every maximal subgroup of 𝑃 is normal in 𝑁_𝐺(𝑃)=𝑃, and thus they are all 𝑆𝐸-supplemented in 𝑁_𝐺(𝑃). However, 𝐺 is not 2-nilpotent.

Remark 12. The hypothesis in Theorem 9 that 𝑃^′ is 𝑆-quasinormal in 𝐺 still cannot be left out when 𝐺 is solvable and 𝑝 an odd prime number.

Example 13. Let 𝐻 be the elementary abelian 3-group of order 27. Then 𝐻=𝑍₃×𝑍₃×𝑍₃. It is easy to see that a subgroup of Aut(𝐻) is isomorphic to 𝑍₁₃⋊𝑍₃. Now assume that 𝐺=(𝑍₃×𝑍₃×𝑍₃)⋊(𝑍₁₃⋊𝑍₃). Let 𝑃₃ be a Sylow 3-subgroup of 𝐺. It is clear that 𝑁_𝐺(𝑃₃)=𝑃₃ and thus every maximal subgroup of 𝑃₃ is 𝑆𝐸-supplemented in 𝑁_𝐺(𝑃₃)=𝑃₃. However, 𝐺 is not 3-nilpotent.

Theorem 14. Let 𝐺 be a group, 𝐻 a normal subgroup of 𝐺 such that 𝐺/𝐻 is 𝑝-nilpotent, and 𝑃 a Sylow 𝑝-subgroup of 𝐻, where 𝑝 is a prime dividing the order of 𝐺 with (|𝐺|,𝑝−1)=1. If there exists a Sylow 𝑝-subgroup 𝑃 of 𝐻 such that every maximal subgroup of 𝑃 is 𝑆𝐸-supplemented in 𝑁_𝐺(𝑃) and such that 𝑃^′ is 𝑆-quasinormal in 𝐺, then 𝐺 is p-nilpotent.

Proof. Assume that the result is not true and let 𝐺 with subgroup 𝐻 be a minimal counterexample to the theorem in respect to |𝐺|+|𝐻|. By Lemmas 2 and 4, we can see that every maximal subgroup of 𝑃 is 𝑆𝐸-supplemented in 𝑁_𝐻(𝑃) and 𝑃^′ is 𝑆-quasinormal in 𝐻. It follows that 𝐻 is 𝑝-nilpotent by Theorem 9. Let 𝑀 be the normal 𝑝-complement of 𝐻; then 𝑀⊴𝐺. If 𝑀≠1, we consider the factor group 𝐺/𝑀 with subgroup 𝐻/𝑀. It is clear that 𝐻=𝑃𝑀 and (|𝑃|,|𝑀|)=1. With a similar argument as in step (4) of Theorem 9, we obtain that the hypothesis still holds for 𝐺/𝑀 with subgroup 𝐻/𝑀. Thereby the choice of 𝐺 implies that 𝐺/𝑀 is 𝑝-nilpotent. Consequently, 𝐺 is 𝑝-nilpotent, which is a contradiction. Now we may suppose that 𝑀=1; that is, 𝐻=𝑃 is a 𝑝-group. Let 𝑇/𝑃 be the normal 𝑝-complement of 𝐺/𝑃; this makes sense as 𝐺/𝑃=𝐺/𝐻 is 𝑝-nilpotent. It is easy to see that every maximal subgroup of 𝑃 is 𝑆𝐸-supplemented in 𝑁_𝑇(𝑃) and 𝑃^′ is 𝑆-quasinormal in 𝑇, whence 𝑇 is 𝑝-nilpotent by Theorem 9. As a result, 𝑇_𝑝′ char 𝑇⊴𝐺 implying that 𝑇_𝑝′ is also a normal Hall 𝑝^′-subgroup of 𝐺; that is, 𝐺 is 𝑝-nilpotent, a contradiction. The proof of the theorem is now complete.

The following result now follows directly from Theorem 14.

Corollary 15. Let 𝐺 be a group and assume 𝑝 is a prime dividing the order of 𝐺 with (|𝐺|,𝑝−1)=1. Suppose that 𝐻 is a normal subgroup of 𝐺 such that 𝐺/𝐻 is 𝑝-nilpotent. If there exists a Sylow 𝑝-subgroup 𝑃 of  𝐻 such that every maximal subgroup of 𝑃 is 𝑆-quasinormally embedded or weakly 𝑆𝐸-supplemented in 𝑁_𝐺(𝑃) and such that 𝑃^′ is 𝑆-quasinormal in 𝐺, then 𝐺 is 𝑝-nilpotent.

The following two theorems study the supersolvability of a group.

Theorem 16. Let 𝐺 be a group. Suppose that for any prime 𝑝 dividing |𝐺|, there exists a Sylow 𝑝-subgroup 𝑃 of 𝐺 such that every maximal subgroup of 𝑃 is 𝑆𝐸-supplemented in 𝑁_𝐺(𝑃) and such that 𝑃^′ is 𝑆-quasinormal in 𝐺. Then 𝐺 is supersolvable.

Proof. Let 𝐺 be a minimal counterexample. According to Theorem 9, it is easy to see that 𝐺 is 𝑝-nilpotent for the minimal prime 𝑝 dividing |𝐺|. Let 𝐾 be the normal 𝑝-complement of 𝐺. Then by Lemmas 2 and 4, it is clear that 𝐾 meets the hypothesis of the theorem, and then 𝐾 is supersolvable by the choice of 𝐺. Let 𝑞= max 𝜋(𝐾) and 𝑄∈Syl_𝑞(𝐾). Then 𝑄∈Syl_𝑞(𝐺) and 𝑄⊴𝐺. Let 𝑁 be a minimal normal subgroup of 𝐺 contained in 𝑄. Thus 𝑁 is an elementary abelian 𝑞-subgroup of 𝐺. Now we consider the factor group 𝐺/𝑁. By Lemmas 2 and 4, we know that every maximal subgroup 𝑄₁/𝑁 of 𝑄/𝑁 is 𝑆𝐸-supplemented in 𝑁_𝐺/𝑁(𝑄/𝑁)=𝑁_𝐺(𝑄)/𝑁=𝐺/𝑁 and (𝑄/𝑁)^′=𝑄^′𝑁/𝑁 is 𝑆-quasinormal in 𝐺/𝑁. Let 𝑅𝑁/𝑁∈Syl_𝑟(𝐺/𝑁) for any 𝑟≠𝑞. Then for any maximal subgroup 𝑇/𝑁 of 𝑅𝑁/𝑁, we know that 𝑇=𝑅₁𝑁, where 𝑅₁ is a maximal subgroup of 𝑅. From the hypothesis, 𝑅₁ is 𝑆𝐸-supplemented in 𝑁_𝐺(𝑅). Then there is a subgroup 𝐾 of 𝑁_𝐺(𝑅) such that 𝑁_𝐺(𝑅)=𝑅₁𝐾 and 𝑅₁∩𝐾≤(𝑅₁)_{𝑠𝑒𝑁𝐺(𝑅)}. Because 𝑅∈Syl_𝑟(𝐺), we know that 𝑁_𝐺/𝑁𝑅𝑁𝑁=𝑁_𝐺(𝑅)𝑁𝑁=𝑅₁𝑁𝑁𝐾𝑁𝑁. (4) Since (|𝑅₁|,|𝑁|)=1, 𝑅₁=𝑅∩𝐾𝑁¹⋅|𝐾𝑁|_𝑟𝑅₁𝐾𝑁_𝑟=𝑅₁⋅|𝐾|_𝑟𝑁_𝐺(𝑅)𝑁_𝑟=𝑅₁⋅|𝐾|_𝑟𝑁_𝐺(𝑅)_𝑟=𝑅₁∩𝐾. (5) This means that 𝑅₁∩𝐾𝑁=𝑅₁∩𝐾; thus 𝑅₁𝑁𝑁∩𝐾𝑁𝑁=𝑅₁𝑁∩𝐾𝑁𝑁=𝑅₁∩𝐾𝑁𝑁𝑁=𝑅₁∩𝐾𝑁𝑁≤𝑅_{1𝑠𝑒𝑁𝐺(𝑅)}𝑁𝑁. (6) Note that (𝑅₁)_{𝑠𝑒𝑁𝐺(𝑅)}𝑁/𝑁=(𝑅₁𝑁/𝑁)_{𝑠𝑒(𝑁𝐺/𝑁(𝑅𝑁/𝑁))} by Lemma 3. Consequently, 𝐺/𝑁 satisfies the hypothesis of the theorem. It follows that the choice of 𝐺 implies that 𝐺/𝑁 is supersolvable. As the class of all supersolvable subgroups is a saturated formation, we may assume that 𝑁 is the unique minimal normal subgroup of 𝐺 contained in 𝑄 with 𝑁≰Φ(𝐺), so there exists a maximal subgroup 𝑀 of 𝐺 such that 𝐺=𝑀𝑁 and 𝑀∩𝑁=1. Since 𝑄=𝑂_𝑞(𝐺)≤𝐹(𝐺)≤𝐶_𝐺(𝑁),  𝑁 and 𝑀 normalize 𝑄∩𝑀, whence 𝑄∩𝑀⊴𝐺. Thus 𝑄∩𝑀=1 or 𝑁≤𝑄∩𝑀. If the latter case holds, then 𝑁≤𝑀; namely, 𝐺=𝑁𝑀=𝑀, a contradiction. Hence 𝑄∩𝑀=1 and 𝑄=𝑁 is a minimal normal subgroup of 𝐺.
Let 𝑄₁ be a maximal subgroup of 𝑄 and 𝑇 be a supplement of 𝑄₁ in 𝐺; then 𝑄₁𝑇=𝐺 and so 𝑄=𝑄∩𝑄₁𝑇=𝑄₁(𝑄∩𝑇). This means that 𝑄∩𝑇=1. However, since 𝑄∩𝑇 is normal in 𝐺 and 𝑄 is a minimal normal subgroup of 𝐺, we have 𝑄∩𝑇=𝑄. Hence 𝑇=𝐺 is the unique supplement of 𝑄₁ in 𝐺. Then 𝑄₁∩𝑇≤(𝑄₁)_𝑠𝐺 by Lemma 6; namely, (𝑄₁)_𝑠𝐺=𝑄₁ is 𝑆-quasinormal in 𝐺. Thus 𝑁_𝐺(𝑄₁)≥𝑂^𝑞(𝐺) by Lemma 5. Since 𝑄₁⊲𝑄, we have 𝑄₁⊲𝑄𝑂^𝑞(𝐺)=𝐺. It follows from the minimal normality of 𝑄 in 𝐺 that 𝑄₁=1 and so |𝑄|=𝑞. Since 𝐺/𝑄 is supersolvable, 𝐺 is supersolvable, a contradiction. The proof of the theorem is now complete.

Theorem 17. Let ℱ be a saturated formation containing the class of all supersolvable groups 𝒰, and assume that 𝐺 is a group with a normal subgroup 𝐻 satisfying 𝐺/𝐻∈ℱ. Suppose that for any prime 𝑝 dividing |𝐻|, there exists a Sylow 𝑝-subgroup 𝑃 of 𝐻 such that every maximal subgroup of 𝑃 is 𝑆𝐸-supplemented in 𝑁_𝐺(𝑃) and such that also 𝑃^′ is 𝑆-quasinormal in 𝐺; then 𝐺∈ℱ.

Proof. Suppose that the result is not true and let 𝐺 with subgroup 𝐻 be a minimal counterexample to the theorem in respect to |𝐺|+|𝐻|. By Lemmas 2 and 4, it is clear that for any prime 𝑝 dividing |𝐻|, there exists a Sylow 𝑝-subgroup 𝑃 of 𝐻 such that every maximal subgroup of 𝑃 is 𝑆𝐸-supplemented in 𝑁_𝐻(𝑃) and that 𝑃^′ is 𝑆-quasinormal in 𝐻. Then 𝐻 meets the hypothesis of Theorem 16, and thus 𝐻 is supersolvable. Let 𝑞= max 𝜋(𝐻) and 𝑄∈Syl_𝑞(𝐻); then 𝑄⊴𝐺. Let 𝑁 be a minimal normal subgroup of 𝐺 contained in 𝑄, and consider the quotient group 𝐺/𝑁. First, (𝐺/𝑁)/(𝐻/𝑁)=𝐺/𝐻∈ℱ. With a similar argument as in the proof of Theorem 16, we can obtain that 𝐺/𝑁 with subgroup 𝐻/𝑁 meets the hypothesis; hence 𝐺/𝑁∈ℱ by the choice of 𝐺. Consequently, we may assume that 𝐻=𝑄=𝑁 is a minimal normal subgroup of 𝐺. Since ℱ constitutes a saturated formation, 𝑁≰Φ(𝐺). Then there exists a maximal subgroup 𝑀 of 𝐺 such that 𝐺=𝑁𝑀 and 𝑁∩𝑀=1. Let 𝑀_𝑞 be a Sylow 𝑞-subgroup of 𝑀. Then 𝐺_𝑞=𝑄𝑀_𝑞 is a Sylow 𝑞-subgroup of 𝐺. Let 𝑄₁=𝑄∩𝑄₀, where 𝑄₀ is a maximal subgroup of 𝐺_𝑞 containing 𝑀_𝑞. Then 𝑄₁ is a maximal subgroup of 𝑄 and 𝑄₁⊴𝐺_𝑞. From this hypothesis, 𝑄₁ is 𝑆𝐸-supplemented in 𝑁_𝐺(𝑄)=𝐺. Let 𝑇 be any supplement of 𝑄₁ in 𝐺; then 𝑄₁𝑇=𝐺 and 𝑄=𝑄∩𝑄₁𝑇=𝑄₁(𝑄∩𝑇). This means that 𝑄∩𝑇≠1. We note that 𝑄∩𝑇 is normal in 𝐺 and 𝑄 is a minimal normal subgroup of 𝐺, 𝑄∩𝑇=𝑄. thus 𝑇=𝐺 is the unique supplement of 𝑄₁ in 𝐺. As a result, we know that 𝑄₁=𝑄₁∩𝑇≤(𝑄₁)_𝑠𝐺 by Lemma 6. Then (𝑄₁)_𝑠𝐺=𝑄₁ is 𝑆-quasinormal in 𝐺. It follows that 𝑁_𝐺(𝑄₁)≥𝑂^𝑞(𝐺) by Lemma 5. As 𝑄₁⊴𝐺_𝑞, it is easy to see that 𝑁_𝐺(𝑄₁)≥𝐺_𝑞𝑂^𝑞(𝐺)=𝐺. By the minimal normality of 𝑄 in 𝐺, it is clear that 𝑄₁=1 and thus 𝑄 is a cyclic group of order 𝑞. It follows that 𝐺∈ℱ by [6, Lemma 2.16], a contradiction. The proof is completed.

From Theorem 17 the following corollary is immediate.

Corollary 18. Let ℱ be a saturated formation containing the class of all supersolvable groups 𝒰, and let 𝐺 be a group with a normal subgroup 𝐻 satisfying 𝐺/𝐻∈ℱ. If for any prime 𝑝 dividing |𝐻|, there exists a Sylow 𝑝-subgroup 𝑃 of 𝐻 such that every maximal subgroup of 𝑃 is 𝑆-quasinormally embedded or weakly 𝑆𝐸-supplemented in 𝑁_𝐺(𝑃) and such that also 𝑃^′ is 𝑆-quasinormal in 𝐺, then 𝐺∈ℱ.

Acknowledgments

The authors are grateful to the referee for his or her helpful report. This research was supported by NSF of China (10961007, 11161006), NSF of Guangxi (0991101, 0991102), and 2011 higher school personnel subsidy scheme of Guangxi (5070).