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Finite group with some c#-normal and S-quasinormally embedded subgroups

  • Ning Li , Jing Jiang and Jianjun Liu EMAIL logo
Published/Copyright: April 24, 2025
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Open Mathematics
From the journal Open Mathematics Volume 23 Issue 1

Abstract

Let p be a prime that divides the order of a finite group G , and let P be a Sylow p -subgroup of G . Assume that d is the smallest number of generators of P and define ℳ d ( P ) = { P 1 , P 2 , … , P d } as a collection of maximal subgroups of P such that ⋂ i = 1 d P i = Φ ( P ) , the Frattini subgroup of P . This article focuses on the exploration of the structure of a finite group G for which every element of ℳ d ( P ) is either c # -normal or S -quasinormally embedded in G . Our results improve and generalize many known results.

Keywords: finite groups; c #-normal subgroups; S-quasinormally embedded subgroups; supersolvable groups
MSC 2010: 20D10; 20D20

1 Introduction

All groups are assumed to be finite. Our notation and terminology are standard (see, e.g., Robinson [1]).

Following Gaschütz [2], a CAP-subgroup of a group G is termed as a subgroup K of G , provided that K either covers or avoids every chief factor A ⁄ B within G , namely, either K A = K B or K ∩ A = K ∩ B . As a generalization of CAP-subgroup, Wang and Wei [3] introduced the concept of c # -normal subgroup. A subgroup H of a group G is said to be a c # -normal subgroup of G , provided that there exists a normal subgroup M of G satisfying the conditions G = H M and H ∩ M being a CAP-subgroup of G [3, Definition 2.3]. They investigated the impact of c # -normality of certain subgroups of prime power order on the p -supersolvability and p -nilpotency of a group. They proved the following: let N be normal subgroup of a p -solvable group G with G ⁄ N being p -supersolvable. If every maximal subgroup of a Sylow p -subgroup P of G is c # -normal in G , then G itself is p -supersolvable [4]. Furthermore, they also considered the special case, and they obtained the following result: consider a normal subgroup N of a group G with G ⁄ N being p -nilpotent, and denote by M a Sylow p -subgroup of N , where p is the smallest prime factor of dividing the order of G . Suppose that every maximal subgroup of M is a c # -normal subgroup of G , then G is p -nilpotent. It has been proved that the c # -normal subgroups provided better tools for us to study the structure of finite groups (e.g., [3–5]).

A subgroup L of a group G is termed S -quasinormal in G whenever it permutes with all Sylow subgroups of G . Since Kegel [6] introduced the concept of S -quasinormal subgroup, there has been much interest in investigating the topic on the S -quasinormality. In 1998, Ballester-Bolinches and Pedraza-Aguilera [7] introduced the concept of S -quasinormally embedded subgroups as a generalization of S -quasinormal subgroups. A subgroup L of a group G is termed S -quasinormally embedded in G if each Sylow subgroup of L is also a Sylow subgroup of an S -quasinormal subgroup of G [7, Definition]. It is proved in [7] that a group G is supersolvable if all of its Sylow subgroups’ maximal subgroups are S -quasinormally embedded in G . Motivated by the result, Asaad and Heliel [8] showed that a group G is a p -nilpotent group if and only if each maximal subgroup of its Sylow p -subgroup is S -quasinormally embedded in G , where p is the smallest prime dividing ∣ G ∣ . These results have been generalized in several studies such as [7–13]. We note that S -quasinormally embedded subgroups and c # -normal subgroups are two different concepts.

Example 1.1

Let G = A 5 be the alternating group of degree 5, and let X be a Sylow subgroup of G . It is evident that X is an S -quasinormally embedded in G . We can see that G = X G and X ∩ G = X . But X neither covers G ⁄ 1 nor avoids G ⁄ 1 . Thus, X is not a c # -normal subgroup of G .

Let G = S 4 be the symmetric group of degree 4 and X = ⟨ ( 12 ) ⟩ . Then, G = X A 4 and so X ∩ A 4 = 1 , which implies that X is c # -normal in G . But X is not S -quasinormally embedded in G .

It is a natural question to ask how much information about the structure of a finite group we can obtain when a small quantity of c # -normal or S -quasinormally embedded maximal subgroups of Sylow subgroups. In order to use fewer c # -normal or S -quasinormally embedded subgroups to characterize the structure of a finite group G , we employ the following definition (refer to [14]).

Definition 1.2

Let P be a p -group with the smallest generator number d , and denote by ℳ d ( P ) a collection of maximal subgroups { P 1 , P 2 , … , P d } of P , which satisfies ⋂ i = 1 d P i = Φ ( P ) , the Frattini subgroup of P .

For a given P , it is evident that ℳ d ( P ) is not uniquely determined. We know that P contains p d − 1 p − 1 maximal subgroups and

lim d → ∞ ( p d − 1 ) ⁄ ( p − 1 ) d = ∞ ,

so

p d − 1 p − 1 ≫ ∣ ℳ d ( P ) ∣ = d .

In this article, we use the c # -normality or S -quasinormally embedding of maximal subgroups of Sylow subgroup P in ℳ d ( P ) to characterize the structure of a group G . We obtain results concerning the p -supersolvability, p -nilpotency, and supersolvability of G and generalize many known results.

2 Preliminaries

In this section, we show some lemmas, which are required in the proofs of our main results.

Lemma 2.1

[3, Lemma 2.5] Let N be a normal subgroup of a group G, and let H be a c # -normal subgroup of G. Then, H N / N is a c # -normal subgroup of G ⁄ N if one of the following holds:

  1. N ≤ H .

  2. ( ∣ H ∣ , ∣ N ∣ ) = 1 , where ( − , − ) denotes the greatest common divisor.

Lemma 2.2

[7, Lemma 1] Suppose that U is an S-quasinormally embedded subgroup of a group G, and K is a normal subgroup of G . Then,

  1. For any subgroup H of G such that U ≤ H ≤ G , U is S-quasinormally embedded in H .

  2. The subgroup UK is S-quasinormally embedded in G and the quotient group U K ⁄ K is S -quasinormally embedded in G ⁄ K .

The following lemmas are related to S -quasinormal subgroups.

Lemma 2.3

[15, Theorem 1] Given that H is an S -quasinormal subgroup of a group G, the quotient group H ⁄ H G is nilpotent.

Lemma 2.4

[16, Proposition B] For a nilpotent subgroup H a group G, the equivalence of the following two statements holds:

  1. H is S-quasinormal within G.

  2. The Sylow subgroups of H are S-quasinormal within G.

Lemma 2.5

[8, Lemma 2.6] Let G p be a Sylow p-subgroup of a group G, and let P be a maximal subgroup of G p . The equivalence of the following two statements holds:

  1. P is normal within G .

  2. P is S-quasinormal within G .

Lemma 2.6

[6] and [16] Let H be an S-quasinormal subgroup of a group G. Then,

  1. If K is an S-quasinormal subgroup of G, then H ∩ K is also an S -quasinormal subgroup of G .

  2. If H is a p-subgroup, then O p ( G ) ≤ N G ( H ) .

Lemma 2.7

[17] Given that P is a Sylow p-subgroup of a group G and N is a normal subgroup of G with P ∩ N ≤ Φ ( P ) , it follows that N is p-nilpotent.

Lemma 2.8

[18, I, Satz 17.4] Let N be a normal abelian subgroup of a group G. Assume that N ≤ M ≤ G , where ∣ N ∣ and the index of M in G are coprime. If N is complemented in M, then N is complemented in G.

Lemma 2.9

[19, Lemma 2.6] Assume that N is a non-trivial solvable normal subgroup of a group G. If every minimal normal subgroup of G that is contained in N is not contained in Φ ( G ) , then the Fitting subgroup F ( N ) of N is given by the direct product of those minimal normal subgroups of G contained in N .

Lemma 2.10

[1, Theorem 9.3.1] Assume G is a π -separable group and O π ′ ( G ) = 1 , and it follows that C G ( O π ( G ) ) ≤ O π ( G ) .

Lemma 2.11

[20, Lemma 2.8] Suppose G is a group and p is a prime number that divides the order of ∣ G ∣ such that ( ∣ G ∣ , p − 1 ) = 1 :

  1. If N is normal in G of order p, then N ≤ Z ( G ) .

  2. If G has cyclic Sylow p-subgroup, then G is p-nilpotent.

  3. If M ≤ G and ∣ G : M ∣ = p , then M ⊴ G .

3 Main results

In this section, we first study the p -supersolvability of a group G when all members of ℳ d ( P ) assumed to be c # -normal or S -quasinormally embedded in G .

Theorem 3.1

Assume G is a group that is p-solvable, and denote by P a Sylow p-subgroup of G for a prime p dividing the order of G. Then, G is p-supersolvable if and only if all members in some fixed ℳ d ( P ) are either c # -normal or S-quasinormally embedded in G.

Proof

We only need to prove the sufficiency by [4, Theorem 3.1]. Assume the theorem to be false, and consider G as a counterexample of minimal order. Let

ℳ d ( P ) = { P 1 , P 2 , … , P d } .

By our hypothesis, each P i of ℳ d ( P ) is either c # -normal or S -quasinormally embedded in G , where i = 1 , … , d . Without loss of generality, we assume that there exists an integer k with 1 ≤ k ≤ d such that for every m with 1 ≤ m ≤ k , P m is c # -normal in G , and for every n with k + 1 ≤ n ≤ d , P n is S -quasinormally embedded in G . We split the proof into the following steps:

(1) O p ′ ( G ) = 1 .

Otherwise, O p ′ ( G ) ≠ 1 . Since P O p ′ ( G ) ⁄ O p ′ ( G ) is a Sylow p -subgroup of G ⁄ O p ′ ( G ) and P O p ′ ( G ) ⁄ O p ′ ( G ) ≅ P , we can see that P O p ′ ( G ) ⁄ O p ′ ( G ) has the same smallest generator number d as P . Set

ℳ d ( P O p ′ ( G ) ⁄ O p ′ ( G ) ) = { P 1 O p ′ ( G ) ⁄ O p ′ ( G ) , … , P d O p ′ ( G ) ⁄ O p ′ ( G ) } .

Then, each P i O p ′ ( G ) ⁄ O p ′ ( G ) for i ∈ { 1 , … , d } is either c # -normal or S -quasinormally embedded in G ⁄ O p ′ ( G ) by Lemmas 2.1 and 2.2. Consequently, G ⁄ O p ′ ( G ) fulfills the requirements stipulated in our theorem. Given the minimality of G , it is compelled that G ⁄ O p ′ ( G ) be p -supersolvable, which in turn implies that G must be p -supersolvable, leading to a contradiction.

(2) Φ ( P ) G = 1 , in particular, Φ ( O p ( G ) ) = 1 .

Suppose that Φ ( P ) G ≠ 1 . Then, it is clear that

{ P 1 / Φ ( P ) G , … , P d / Φ ( P ) G } = ℳ d ( P / Φ ( P ) G ) .

Also, each P i / Φ ( P ) G for i ∈ { 1 , … , d } is either c # -normal or S -quasinormally embedded in G / Φ ( P ) G by Lemmas 2.1 and 2.2. Thus, the hypothesis of our theorem is automatically satisfied for G / Φ ( P ) G . So G / Φ ( P ) G is p -supersolvable by the minimality of G . By [18, III, Satz 3.3], we have that Φ ( P ) G ≤ Φ ( G ) and so G is p -supersolvable, which contradicts the minimality of G .

(3) Any minimal normal subgroup of G that is contained within O p ( G ) possesses an order of p .

By statement (1) and the p -solvability of G , it follows that O p ( G ) > 1 . Consider N to be a minimal normal subgroup of G that is contained within O p ( G ) . By hypotheses, every P m is c # -normal in G , and hence, there exists a normal subgroup K m of G such that G = P m K m and P m ∩ K m is a CAP-subgroup of G for all m ∈ { 1 , … , k } , i.e., P m ∩ K m covers or avoids N ⁄ 1 . Suppose that there exists some P m such that P m ∩ K m avoids N ⁄ 1 . Then, P m ∩ K m ∩ N = 1 . By the minimal normality of N , we can see that either N ∩ K m = 1 or N ∩ K m = N . If N ∩ K m = 1 , then N K m / K m is a minimal normal subgroup of G ⁄ K m . But G ⁄ K m is a p -group as G = P m K m , which means that N ≅ N K m / K m is of order p . If N ∩ K m = N , then N ∩ P m = 1 . This derives that P = P m × N , and thus, ∣ N ∣ = p . Now we may assume that every P m ∩ K m covers N ⁄ 1 for all m ∈ { 1 , … , k } . Then, N ≤ P m ∩ K m and so

N ≤ ⋂ m = 1 k ( P m ) G ,

where ( P m ) G is the core of P m in G .

By our assumption, for each n ∈ { k + 1 , … , d } , P n is S -quasinormally embedded in G , and consequently, there exists an S -quasinormal subgroup M n of G for which P n serves as a Sylow p -subgroup. Therefore, we may apply Lemmas 2.3 and 2.4 to see that M n / ( M n ) G is nilpotent and all Sylow subgroups of M n / ( M n ) G are S -quasinormal in G ⁄ ( M n ) G . In particular, P n ( M n ) G / ( M n ) G is S -quasinormal in G / ( M n ) G . It follows from Lemma 2.5 that P n ( M n ) G / ( M n ) G is normal in G / ( M n ) G . We obtain that P n ≤ ( M n ) G and so P ∩ ( M n ) G = P n .

Let

T = ⋂ m = 1 k ( P m ) G ∩ ⋂ n = k + 1 d ( M n ) G ,

then T ⊴ G . By the minimal normality of N , we have that N ∩ ( M n ) G = N or 1. If N ∩ ( M n ) G = N for all n ∈ { k + 1 , … , d } , then N ≤ ( M n ) G , and consequently, N ≤ T . We obtain that

T ∩ P = ⋂ m = 1 k ( P m ) G ∩ ⋂ n = k + 1 d ( M n ) G ∩ P = ⋂ m = 1 k ( P m ) G ∩ ⋂ n = k + 1 d ( M n ) G ∩ P = ⋂ m = 1 k ( P m ) G ∩ ⋂ n = k + 1 d P n ≤ ⋂ i = 1 d P i = Φ ( P ) .

From Lemma 2.7, we deduce that T is p -nilpotent, and hence, by statement (1), T is a p -group. Statement (2) implies that T = N = 1 , which is a contradiction. Therefore, there exists some M i such that N ∩ ( M i ) G = 1 , where i ∈ { k + 1 , … , d } . Therefore, ∣ N ∣ = p .

(4) No counterexample exists.

Consider N as a minimal normal subgroup of G such that N ≤ O p ( G ) . Then, statement (3) and Lemma 2.8 are combined to give that N is complemented in G . We obtain that N ∩ Φ ( G ) = 1 and so O p ( G ) ∩ Φ ( G ) = 1 . We apply Lemma 2.9 to give that

O p ( G ) = N 1 × N 2 × … × N s ,

where each N i (for i = 1 , … , s ) is a minimal normal subgroup of G of order p . Since

G / C G ( N i ) ≅ Aut ( N i )

and Aut ( N i ) is a cyclic group of order p − 1 , we conclude that

G / C G ( O p ( G ) ) = G / ⋂ i = 1 r C G ( N i )

is p -supersolvable. Given that G is p -solvable and O p ′ ( G ) = 1 , it follows that

C G ( O p ( G ) ) ≤ O p ( G )

by Lemma 2.10. It follows that G ⁄ O p ( G ) is p -supersolvable. Now, every chief factor of G contained in O p ( G ) is of order p ; and hence, every p -chief factor of G is cyclic. It follows that G is p -supersolvable, which is a final contradiction.□

As immediate consequences of Theorem 3.1, we have the following.

Corollary 3.2

[4, Theorem 3.1] Consider G as a p-solvable group, and denote by P a Sylow p-subgroup of G, where p is a prime number that divides the order of G. Then, G is p-supersolvable if and only if all members in some fixed ℳ d ( P ) are c # -normal in G.

Corollary 3.3

Consider G as a p-solvable group, and denote by P a Sylow p-subgroup of G, where p is a prime number that divides the order of G. Then, G is p-supersolvable if and only if all members in some fixed ℳ d ( P ) are S-quasinormally embedded in G.

Remark 3.4

The assumption of “ G is p -solvable” in Theorem 3.1 is indispensable. Indeed, G = A 5 is a counterexample for p = 5 .

Revising the method used in the proof of Theorem 3.1, we can obtain some results. In the following theorem, we replace the condition “ G is p -solvable” in Theorem 3.1 by “ ( ∣ G ∣ , p − 1 ) = 1 .”

Theorem 3.5

Let G be a group, p be a prime divisor of ∣ G ∣ with ( ∣ G ∣ , p − 1 ) = 1 , and P be a Sylow p -subgroup of G. Then, G is a p-nilpotent group if and only if all members in some fixed ℳ d ( P ) are either c # -normal or S-quasinormally embedded in G .

Proof

Applying [4, Theorem 3.2], it is only the sufficiency of the condition that is in doubt. Assume the result is incorrect and consider G as a counterexample possessing the smallest possible order. Let

ℳ d ( P ) = { P 1 , P 2 , … , P d } .

With the same arguments as in the proof of Theorem 3.1, we can obtain the statements (1)–(4).

  1. O p ′ ( G ) = 1 .

  2. Φ ( P ) G = 1 , in particular, Φ ( O p ( G ) ) = 1 .

  3. Let N be a p -group. If N is a minimal normal subgroup of G , then ∣ N ∣ = p .

  4. Any minimal normal subgroup of G is a subgroup of O p ( G ) .

Let N be a minimal normal subgroup of G . Then, statement (1) means that p ∣ ∣ N ∣ . Let P i ∈ ℳ d ( P ) . By hypotheses, P i is either c # -normal or S -quasinormally embedded in G . Given that P i is c # -normal in G , a normal subgroup K of G exists, satisfying G = P i K and P i ∩ K being a CAP-subgroup of G , i.e., P i ∩ K covers or avoids N ⁄ 1 . If P i ∩ K covers N ⁄ 1 , then N ≤ P i and so N ≤ O p ( G ) . If P i ∩ K avoids N ⁄ 1 , then P i ∩ K ∩ N = 1 . By the minimal normality of N , we can see that either N ∩ K = 1 or N ∩ K = N . If N ∩ K = 1 , then N K ⁄ K is a minimal normal subgroup of G ⁄ K . It follows that N has order p and so N ≤ O p ( G ) . If N ∩ K = N , then N ∩ P i = 1 . Consequently, ∣ P i N ∣ p = ∣ P i ∣ ∣ N ∣ p and so ∣ N ∣ p = p . Lemma 2.11 and ( ∣ G ∣ , p − 1 ) = 1 are combined to give that N is p -nilpotent. Thus, N is a p -group by statement (1). Hence, N ≤ O p ( G ) .

Now, we may assume that all members of ℳ d ( P ) are S -quasinormally embedded in G and N p = N ∩ P ≰ Φ ( P ) by Lemma 2.7. Without loss of generality, we assume that N p ≰ P 1 ∈ ℳ d ( P ) . Hence, there exists an S -quasinormal subgroup H of G , for which P 1 is a Sylow p -subgroup. Therefore, P 1 H G / H G is a Sylow p -subgroup of H / H G . By Lemmas 2.3 and 2.4, H / H G is nilpotent and P 1 H G / H G is an S -quasinormal subgroup of H / H G . Since P 1 H G / H G is a maximal subgroup of a Sylow p -subgroup of G / H G , we see that P 1 H G / H G is normal in G / H G by Lemma 2.5. Hence, P 1 H G is normal in G , which implies that P 1 ≤ H G . By the minimal normality of N , we obtain that N ∩ H G = N or 1. If N ∩ H G = N , then N p ≤ P ∩ H G = P 1 , which is a contradiction. Therefore, N ∩ H G = 1 and so N ∩ P 1 = 1 . Consequently, ∣ N ∣ p = p . It means that N is p -nilpotent, and we apply statement (1) to give that N ≤ O p ( G ) .

(5) The final contradiction.

By statements (2) and (4), we obtain that O p ( G ) ≠ 1 and O p ( G ) is elementary abelian. Applying Lemmas 2.8 and 2.9, we can see that

O p ( G ) = N 1 × N 2 × … × N r ,

where N i is a minimal normal subgroup of G with ∣ N i ∣ = p . Furthermore, O p ( G ) has a complement K in G , i.e.,

G = O p ( G ) ⋊ K .

Let

T = ⋂ i = 1 r C G ( N i ) .

Since N i ≤ Z ( P ) , we have P ≤ T . If T ∩ K ≠ 1 , then there exists a minimal normal subgroup L of G such that L ≤ T ∩ K and L ≰ O p ( G ) , which contradicts with statement (4). Hence, T ∩ K = 1 . Moreover, we obtain that P = O p ( G ) . Set

P i = N 1 × N 2 × … × N i − 1 × N i + 1 × … × N r .

Then, P i is normal in G and ∣ G ⁄ P i ∣ p = p . By Lemma 2.11, G ⁄ P i is p -nilpotent. Furthermore,

G / ⋂ i = 1 r P i

is p -nilpotent, but ⋂ i = 1 r P i = 1 . It follows that G is p -nilpotent, which a final contradiction.□

Therefore, from Theorem 3.5, we obtain

Corollary 3.6

[4, Theorem 3.2] and [5, Theorem 3.1] Let p be a prime divisor of the order of a group G with ( ∣ G ∣ , p − 1 ) = 1 , and let P be a Sylow p-subgroup of G. Then, G is a p-nilpotent group if and only if all members in some fixed ℳ d ( P ) are c # -normal in G.

Corollary 3.7

[10, Theorem 3.1] Let p be a prime divisor of the order of a group G with ( ∣ G ∣ , p − 1 ) = 1 , and let P be a Sylow p-subgroup of G. Then, G is a p-nilpotent group if and only if all members in some fixed ℳ d ( P ) are S-quasinormally embedded in G.

Combining Theorems 3.1 and 3.5, we obtain the following result.

Theorem 3.8

Consider an arbitrary prime divisor p of the order of a group, and denote by P a Sylow p-subgroup of G. A group G is supersolvable if and only if all members in some fixed ℳ d ( P ) are either c # -normal or S-quasinormally embedded in G.

Proof

Only the sufficiency needs to be shown. Let p be the smallest prime of the order of G . Then, from Theorem 3.5   G is p -nilpotent. It follows that G is solvable. We apply Theorem 3.1 to see that for arbitrary prime divisor of ∣ G ∣ , G is p -supersolvable, and thus, G is supersolvable.□

In view of Theorem 3.8, we have the following.

Corollary 3.9

[4, Theorem 3.5] A group G is supersolvable if and only if all members in some fixed ℳ d ( P ) are c # -normal in G, for every Sylow subgroup P of G.

Corollary 3.10

[10, Theorem 3.2] A group G is supersolvable if and only if for each Sylow subgroup P of G, all members in some fixed ℳ d ( P ) are S -quasinormally embedded in G.

It is recalled that a formation F is defined as a class of groups that is closed under the operations of taking homomorphic images and subdirect products. A formation F is termed saturated if G ∈ F holds whenever G ⁄ Φ ( G ) ∈ F is satisfied. Let U be the class of all groups that are supersolvable. It is evident that U constitutes a saturated formation. We also discover that aforementioned results cannot be extended to saturated formation, specifically:

Define F as a saturated formation encompassing U . Consider K as a normal subgroup of a group G with the property that G ⁄ K ∈ F . Assume that for every prime p dividing the order of K , P ∈ Syl p ( K ) , and all members in some fixed ℳ d ( P ) are either c # -normal or S -quasinormally embedded in G . But G ∉ F .

Example 3.11

The formation function, denoted by f ( p ) , is defined as follows:

f ( p ) = { the class of p ′ -groups for any prime p } .

Furthermore, let F be the locally defined formation based on the set { f ( p ) } . Suppose that K ⁄ L is a p -chief factor of a supersolvable group X , then X / C X ( K ⁄ L ) is a cyclic group of order dividing p − 1 and so X / C X ( K ⁄ L ) ∈ f ( p ) . It follows that X ∈ F , and hence, U ⊆ F . The inclusion of A 4 ∈ F is evident.

Let P = ⟨ a , b , c ⟩ be an elementary abelian group of order 3 3 , and let u , v ∈ Aut ( P ) such that

u = a b c c a b and v = a b c b c − 1 a − 1 .

Then, u 3 = v 3 = ( u v ) 2 = 1 , which means that A = ⟨ u , v ⟩ ≅ A 4 . Set G = P ⋊ A , the semidirect product of P and A . We can see that P is an irreducible and faithful A 4 -module over G F ( p ) , implying that P is a minimal normal subgroup of G with C A ( P ) = 1 . Given that A 4 ∈ F and G ⁄ P ≅ A ≅ A 4 , we conclude that G ⁄ P ∈ F . Define M = P S , with S is a Sylow 2-subgroup of G . Note that O 3 ( G ) ≤ M ⊴ G . Since S is an elementary abelian group of order 4, we derive that any minimal normal subgroup of M contained in P has order 3. Invoking Maschke’s theorem [1, Theorem 8.1.2], we deduce that P is a completely reducible S -module. Consequently, P admits a decomposition as P = ⟨ a 1 ⟩ × ⟨ a 2 ⟩ × ⟨ a 3 ⟩ , with ⟨ a i ⟩ (i = 1, 2, 3) being S -invariant. Let P i = ⟨ a j ∣ j ≠ i ⟩ . Then,

ℳ d ( P ) = { P 1 , P 2 , P 3 } ,

and each P i in ℳ d ( P ) is S -quasinormally embedded within G . Furthermore, P is identified as a 3-chief factor of G and G / C G ( P ) ≅ A 4 , which is not a 3 ′ -group. Thus, G ∉ F .

Acknowledgements

The authors would like to thank the referees for their valuable suggestions and useful comments that contributed to the final version of this manuscript.

  1. Funding information: This work was supported by the National Natural Science Foundation of China (Grant No. 12471019).

  2. Author contributions: All authors have accepted responsibility for the entire content of this manuscript and consented to its submission to the journal, reviewed all the results, and approved the final version of the manuscript.

  3. Conflict of interest: The authors state no conflict of interest.

  4. Data availability statement: No data were used for the research described in the article.

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Received: 2024-06-26
Revised: 2024-12-11
Accepted: 2025-02-13
Published Online: 2025-04-24

© 2025 the author(s), published by De Gruyter

This work is licensed under the Creative Commons Attribution 4.0 International License.

Articles in the same Issue

  1. Special Issue on Contemporary Developments in Graphs Defined on Algebraic Structures
  2. Forbidden subgraphs of TI-power graphs of finite groups
  3. Finite group with some c#-normal and S-quasinormally embedded subgroups
  4. Classifying cubic symmetric graphs of order 88p and 88p 2
  5. Simplicial complexes defined on groups
  6. Two-sided zero-divisor graphs of orientation-preserving and order-decreasing transformation semigroups
  7. Further results on permanents of Laplacian matrices of trees
  8. Research Articles
  9. Dynamics of particulate emissions in the presence of autonomous vehicles
  10. The regularity of solutions to the Lp Gauss image problem
  11. Exploring homotopy with hyperspherical tracking to find complex roots with application to electrical circuits
  12. The ill-posedness of the (non-)periodic traveling wave solution for the deformed continuous Heisenberg spin equation
  13. Some results on value distribution concerning Hayman's alternative
  14. 𝕮-inverse of graphs and mixed graphs
  15. A note on the global existence and boundedness of an N-dimensional parabolic-elliptic predator-prey system with indirect pursuit-evasion interaction
  16. On a question of permutation groups acting on the power set
  17. Chebyshev polynomials of the first kind and the univariate Lommel function: Integral representations
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  19. Spectrum boundary domination of semiregularities in Banach algebras
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  69. Subharmonic functions and associated measures in ℝn
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  72. Multiple positive solutions to a p-Kirchhoff equation with logarithmic terms and concave terms
  73. Joint approximation of analytic functions by the shifts of Hurwitz zeta-functions in short intervals
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  75. Some new Hermite-Hadamard type inequalities for product of strongly h-convex functions on ellipsoids and balls
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  77. On an uncertainty principle for small index subgroups of finite fields
  78. On a generalization of I-regularity
  79. Spin(8, ℂ)-Higgs bundles fixed points through spectral data
  80. Coloring the vertices of a graph with mutual-visibility property
  81. Embedding of lattices and K3-covers of an enriques surface
  82. Algorithm and linear convergence of the H-spectral radius of weakly irreducible quasi-positive tensors
  83. q-Stirling sequence spaces associated with q-Bell numbers
  84. Multiple G-Stratonovich integral in G-expectation space
https://doi.org/10.1515/math-2025-0140
Keywords for this article
finite groups; c #-normal subgroups; S-quasinormally embedded subgroups; supersolvable groups
Creative Commons
BY 4.0

Articles in the same Issue

  1. Special Issue on Contemporary Developments in Graphs Defined on Algebraic Structures
  2. Forbidden subgraphs of TI-power graphs of finite groups
  3. Finite group with some c#-normal and S-quasinormally embedded subgroups
  4. Classifying cubic symmetric graphs of order 88p and 88p 2
  5. Simplicial complexes defined on groups
  6. Two-sided zero-divisor graphs of orientation-preserving and order-decreasing transformation semigroups
  7. Further results on permanents of Laplacian matrices of trees
  8. Research Articles
  9. Dynamics of particulate emissions in the presence of autonomous vehicles
  10. The regularity of solutions to the Lp Gauss image problem
  11. Exploring homotopy with hyperspherical tracking to find complex roots with application to electrical circuits
  12. The ill-posedness of the (non-)periodic traveling wave solution for the deformed continuous Heisenberg spin equation
  13. Some results on value distribution concerning Hayman's alternative
  14. 𝕮-inverse of graphs and mixed graphs
  15. A note on the global existence and boundedness of an N-dimensional parabolic-elliptic predator-prey system with indirect pursuit-evasion interaction
  16. On a question of permutation groups acting on the power set
  17. Chebyshev polynomials of the first kind and the univariate Lommel function: Integral representations
  18. Blow-up of solutions for Euler-Bernoulli equation with nonlinear time delay
  19. Spectrum boundary domination of semiregularities in Banach algebras
  20. Statistical inference and data analysis of the record-based transmuted Burr X model
  21. A modified predictor–corrector scheme with graded mesh for numerical solutions of nonlinear Ψ-caputo fractional-order systems
  22. Dynamical properties of two-diffusion SIR epidemic model with Markovian switching
  23. Classes of modules closed under projective covers
  24. On the dimension of the algebraic sum of subspaces
  25. Periodic or homoclinic orbit bifurcated from a heteroclinic loop for high-dimensional systems
  26. On tangent bundles of Walker four-manifolds
  27. Regularity of weak solutions to the 3D stationary tropical climate model
  28. A new result for entire functions and their shifts with two shared values
  29. Freely quasiconformal and locally weakly quasisymmetric mappings in metric spaces
  30. On the spectral radius and energy of the degree distance matrix of a connected graph
  31. Solving the quartic by conics
  32. A topology related to implication and upsets on a bounded BCK-algebra
  33. On a subclass of multivalent functions defined by generalized multiplier transformation
  34. Local minimizers for the NLS equation with localized nonlinearity on noncompact metric graphs
  35. Approximate multi-Cauchy mappings on certain groupoids
  36. Multiple solutions for a class of fourth-order elliptic equations with critical growth
  37. A note on weighted measure-theoretic pressure
  38. Majorization-type inequalities for (m, M, ψ)-convex functions with applications
  39. Recurrence for probabilistic extension of Dowling polynomials
  40. A generalized fixed-point theorem for set-valued mappings in b-metric spaces
  41. Unraveling chaos: A topological analysis of simplicial homology groups and their foldings
  42. Global existence and blow-up of solutions to pseudo-parabolic equation for Baouendi-Grushin operator
  43. A characterization of the translational hull of a weakly type B semigroup with E-properties
  44. Some new bounds on resolvent energy of a graph
  45. Carmichael numbers composed of Piatetski-Shapiro primes in Beatty sequences
  46. The number of rational points of some classes of algebraic varieties over finite fields
  47. Singular direction of meromorphic functions with finite logarithmic order
  48. Pullback attractors for a class of second-order delay evolution equations with dispersive and dissipative terms on unbounded domain
  49. Eigenfunctions on an infinite Schrödinger network
  50. Boundedness of fractional sublinear operators on weighted grand Herz-Morrey spaces with variable exponents
  51. On SI2-convergence in T0-spaces
  52. Bubbles clustered inside for almost-critical problems
  53. Classification and irreducibility of a class of integer polynomials
  54. Existence and multiplicity of positive solutions for multiparameter periodic systems
  55. Averaging method in optimal control problems for integro-differential equations
  56. On superstability of derivations in Banach algebras
  57. Investigating the modified UO-iteration process in Banach spaces by a digraph
  58. The evaluation of a definite integral by the method of brackets illustrating its flexibility
  59. Existence of positive periodic solutions for evolution equations with delay in ordered Banach spaces
  60. Tilings, sub-tilings, and spectral sets on p-adic space
  61. The higher mapping cone axiom
  62. Continuity and essential norm of operators defined by infinite tridiagonal matrices in weighted Orlicz and l∞ spaces
  63. A family of commuting contraction semigroups on l 1 ( N ) and l ∞ ( N )
  64. Pullback attractor of the 2D non-autonomous magneto-micropolar fluid equations
  65. Maximal function and generalized fractional integral operators on the weighted Orlicz-Lorentz-Morrey spaces
  66. On a nonlinear boundary value problems with impulse action
  67. Normalized ground-states for the Sobolev critical Kirchhoff equation with at least mass critical growth
  68. Decompositions of the extended Selberg class functions
  69. Subharmonic functions and associated measures in ℝn
  70. Some new Fejér type inequalities for (h, g; α - m)-convex functions
  71. The robust isolated calmness of spectral norm regularized convex matrix optimization problems
  72. Multiple positive solutions to a p-Kirchhoff equation with logarithmic terms and concave terms
  73. Joint approximation of analytic functions by the shifts of Hurwitz zeta-functions in short intervals
  74. Green's graphs of a semigroup
  75. Some new Hermite-Hadamard type inequalities for product of strongly h-convex functions on ellipsoids and balls
  76. Infinitely many solutions for a class of Kirchhoff-type equations
  77. On an uncertainty principle for small index subgroups of finite fields
  78. On a generalization of I-regularity
  79. Spin(8, ℂ)-Higgs bundles fixed points through spectral data
  80. Coloring the vertices of a graph with mutual-visibility property
  81. Embedding of lattices and K3-covers of an enriques surface
  82. Algorithm and linear convergence of the H-spectral radius of weakly irreducible quasi-positive tensors
  83. q-Stirling sequence spaces associated with q-Bell numbers
  84. Multiple G-Stratonovich integral in G-expectation space
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