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Some new characterizations of finite p-nilpotent groups

  • Fengyan Xie and Jinbao Li EMAIL logo
Published/Copyright: December 31, 2022
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Open Mathematics
From the journal Open Mathematics Volume 20 Issue 1

Abstract

In this article, some new sufficient conditions of p-nilpotency of finite groups are obtained by using c-normality and Φ-supplementary of the maximal or the 2-maximal subgroups of the Sylow p-subgroups.

Keywords: c-normal subgroups; Φ-supplemented subgroups; 2-maximal subgroups; p-nilpotency
MSC 2010: 20D05; 20D10; 20D20

1 Introduction

All groups considered are finite.

For a group G , we denote by Φ ( G ) the intersection of all maximal subgroups of G . Let H be a subgroup of G . H is said to be supplemented in G provided that there exists a subgroup T of G such that G = H T . The supplemented subgroups have a significant influence on the structure of finite groups. It was proved by Kegel in [1,2] that a group G is soluble if every maximal subgroup of G has a cyclic supplement in G or if some nilpotent subgroup of G has a nilpotent supplement in G . In [3], Wang introduced the concept of c-normality of subgroups. H is said to be c-normal in G [3] if there exists a normal subgroup T of G such that G = H T and H ∩ T ≤ H G . Furthermore, Yu [4] studied the relationship between Φ -supplemented subgroups and the structure of finite groups. We say that H is Φ -supplemented in G [4] if there exists a normal subgroup T of G such that G = H T and H ∩ T ≤ Φ ( H ) . By using these special supplemented subgroups, many authors have obtained a series of interesting results (see [3,4, 5,6,7, 8,9]). We further carried out this study and obtained some new criteria for the p-nilpotency of finite groups in terms of c-normality and Φ -supplementary of the maximal or 2-maximal subgroups of the Sylow p-subgroups.

All other unexplained notions and terminology are standard and the reader is referred to [10].

2 Preliminaries

In this section, we recall some facts, which will be used in this article.

Lemma 2.1

Suppose that H is c-normal in G. Then, the following statements hold:

  1. If H ≤ M ≤ G , then H is c-normal in M .

  2. If N ⊴ G and N ≤ H , then H / N is c-normal in G / N .

  3. If N ⊴ G and ( ∣ H ∣ , ∣ N ∣ ) = 1 , then H N / N is c-normal in G / N .

Proof

See [3, Lemma 2.1].□

Lemma 2.2

Suppose that H is Φ -supplemented in G. Then, the following statements (1)–(3) hold:

  1. If H ≤ M ≤ G , then H is Φ -supplemented in M .

  2. If N ⊴ G and N ≤ H , then H / N is Φ -supplemented in G / N .

  3. If N ⊴ G and ( ∣ H ∣ , ∣ N ∣ ) = 1 , then H N / N is Φ -supplemented in G / N .

Proof

See [4].□

Lemma 2.3

Suppose that N is normal in G and G/N is a p-nilpotent group, where p is a prime divisor of ∣ G ∣ and ( ∣ G ∣ , p − 1 ) = 1 . If ∣ N ∣ = p , then G is p-nilpotent.

Proof

Since ∣ N ∣ = p , ∣ Aut ( N ) ∣ = p − 1 and N ≤ C G ( N ) . Because N is normal in G and ( ∣ G ∣ , p − 1 ) = 1 , ( ∣ N G ( N ) / C G ( N ) ∣ , p − 1 ) = 1 . Since N G ( N ) / C G ( N ) is isomorphic to some subgroup of Aut ( N ) , N G ( N ) = C G ( N ) , that is, N ≤ Z ( G ) . Hence, G is p-nilpotent by G / N is p-nilpotent.□

Lemma 2.4

Let G be A 4 -free and p be prime divisor of ∣ G ∣ with ( ∣ G ∣ , p − 1 ) = 1 . If p 3 ∤ ∣ G ∣ , then G is p-nilpotent.

Proof

See [11, Lemma 2.8].□

Lemma 2.5

If P is a Sylow p-subgroup of G, where p is a prime divisor of ∣ G ∣ , and N ⊴ G such that P ∩ N ≤ Φ ( P ) , then N is p-nilpotent.

Proof

See [10, Chapter 4, Theorem 4.7].□

3 Main results

Theorem 3.1

Suppose that P is a Sylow p-subgroup of a group G, where p is a prime divisor of ∣ G ∣ and ( ∣ G ∣ , p − 1 ) = 1 . If every maximal subgroup of P is c-normal or Φ -supplemented in G, then G is p-nilpotent.

Proof

Suppose that the statement is not true, and let G be a counterexample of minimal order. Then, we have the following steps.

(1) There exists a unique minimal normal subgroup N in G . Moreover, G / N is p-nilpotent.

We pick a minimal normal subgroup of G , say N . Since P is a Sylow p-subgroup of G , P N / N is a Sylow p-subgroup of G / N . Let M / N be a maximal subgroup of P N / N and set H = M ∩ P . Then, M = M ⋂ P N = ( M ∩ P ) N = H N and H ≤ P . Therefore,

∣ P : H ∣ = ∣ P : M ∩ P ∣ = ∣ P ∣ ∣ M ∣ ∣ P ∣ / ∣ M P ∣ = ∣ M P ∣ ∣ M ∣ = ∣ N P / N ∣ ∣ M / N ∣ = p ,

that is, H is a maximal subgroup of P . By the hypotheses, there exists a normal subgroup T of G such that H ∩ T ≤ H G or H ∩ T ≤ Φ ( H ) . Suppose that N is not a subgroup of T . Then, N ∩ T = 1 . Since both N and T are normal in G , ∣ N T ∣ = ∣ N ∣ ∣ T ∣ and ∣ N T ∣ ∣ ∣ G ∣ . Because, ∣ H T / T ∣ = ∣ H / H ∩ T ∣ is a power of p, N is an abelian subgroup of G and N < P . Thus, M is the maximal subgroup of P . By the hypotheses, M is c-normal or Φ -supplemented in G . It follows that M / N is c-normal or Φ -supplemented in G / N by Lemmas 2.1 and 2.2. Now, we assume that N ≤ T . Since G = H T and T is normal in G , G / N = H T / N = ( M / N ) ( T / N ) and T / N is normal in G / N . If H ∩ T ≤ H G , then ( M / N ) ∩ ( T / N ) = ( H N ∩ T ) / N = ( H ∩ T ) N / N ≤ H G N / N ≤ ( H N / N ) ( G / N ) = ( M / N ) ( G / N ) . If H ∩ T ≤ Φ ( H ) , then ( M / N ) ∩ ( T / N ) = ( H N ∩ T ) / N = ( H ∩ T ) N / N ≤ Φ ( H ) N / N ≤ Φ ( H N / N ) = Φ ( M / N ) . Therefore, M / N is c-normal or Φ -supplemented in G / N . Obviously, ( ∣ G / N ∣ , p − 1 ) = 1 . Hence, G / N satisfies the hypotheses. By the choice of G , G / N is p-nilpotent. Because the class of all p-nilpotent groups forms a saturated formation, we deduce that N is the only minimal normal subgroup in G .

(2) N is not p-nilpotent.

Assume that N is p-nilpotent. Let L be the normal p-complement of N . Because L char N and N is normal in G , L is normal in G . The minimal normality of N shows that L = 1 , that is, N is a p-subgroup. Since G / N is p-nilpotent, Φ ( G ) = 1 . Let M be a maximal subgroup of G with G = [ N ] M . Suppose that K is a Sylow p-subgroup of M such that P = [ N ] K . Let A be a maximal subgroup of N and A is normal in P . Set H = A K . Then, H is a maximal subgroup of P . By the hypotheses, there exists a normal subgroup T of G such that H ∩ T ≤ H G or H ∩ T ≤ Φ ( H ) . Because N is the unique minimal normal subgroup of G , N ≤ T . If H ∩ T ≤ Φ ( H ) , then H = H ∩ P = H ∩ N K = ( H ∩ N ) K ≤ ( H ∩ T ) K ≤ Φ ( H ) K . Since H = A K , H = K . Because P = [ N ] H and H is a maximal subgroup of P , ∣ N ∣ = p . By (1) and Lemma 2.3, G is p-nilpotent. This contradiction shows that H ∩ T ≤ H G and H G ≠ 1 . Then, N ≤ H G by (1). Thus, P = [ N ] K ≤ H , a contradiction.

(3) The finial contradiction.

If N P < G , then N P satisfies the hypotheses. The choice of G yields that N P is p-nilpotent, and so N is p-nilpotent, a contradiction by Step (2). Therefore, N P = G . Since G / N = N P / N is p-subgroup, there exists a normal subgroup M / N of G / N such that ∣ G : M ∣ = p . Because P is a Sylow p-subgroup of G , G = P M . Then, ∣ P : P ∩ M ∣ = ∣ P M : M ∣ = p , that is, P ∩ M is a maximal subgroup of P . Set H = P ∩ M . By the hypotheses, there exists a normal subgroup T of G such that G = H T and H ∩ T ≤ H G or H ∩ T ≤ Φ ( H ) . Because N is the unique minimal normal subgroup of G , N ≤ T and N ≤ M . Suppose first that H ∩ T ≤ Φ ( H ) . Since H is normal in P , Φ ( H ) ≤ Φ ( P ) . Then, P ∩ N ≤ P ∩ ( M ∩ T ) = ( P ∩ M ) ∩ T = H ∩ T ≤ Φ ( H ) ≤ Φ ( P ) . By Lemma 2.5, N is p-nilpotent, a contradiction by Step (2). If H ∩ T ≤ H G and H G ≠ 1 , then, N ≤ H G by the unique minimal normality of N . Therefore, N is p-nilpotent. This is the final contradiction and the proof is completed.□

Corollary 3.1

Assume that P is a Sylow p-subgroup of G, where p is the smallest prime divisor of ∣ G ∣ . Suppose that every maximal subgroup of P is c-normal or Φ -supplemented in G . Then, G is p-nilpotent.

Corollary 3.2

Suppose that every maximal subgroup of any Sylow subgroup of a group is c-normal or Φ -supplemented in G. Then, G is a Sylow tower group of supersolvable type.

Proof

Let p be the smallest prime dividing ∣ G ∣ and P be a Sylow p-subgroup of G . By Corollary 3.1, G is p-nilpotent. Let K be the normal p-complement of G . By Lemmas 2.1 and 2.2, K satisfies the hypothesis of the corollary. It follows that K is a Sylow tower group of supersolvable type by induction, which implies that G is also a Sylow tower group of supersolvable type.□

Corollary 3.3

Assume that P is a Sylow p-subgroup of G, where p is a prime divisor of ∣ G ∣ and ( ∣ G ∣ , p − 1 ) = 1 . Suppose that every maximal subgroup of P is c-normal in G. Then, G is p-nilpotent.

Corollary 3.4

Assume that P is a Sylow p-subgroup of G, where p is a prime divisor of ∣ G ∣ and ( ∣ G ∣ , p − 1 ) = 1 . Suppose that every maximal subgroup of P is Φ -supplemented in G . Then, G is p-nilpotent.

Corollary 3.5

Let p be a prime dividing the order of G with ( ∣ G ∣ , p − 1 ) = 1 and E be a normal subgroup of G such that G / E is p-nilpotent. Suppose that P is a Sylow p-subgroup of E and every maximal subgroup of P is c-normal or Φ -supplemented in G. Then, G is p-nilpotent.

Proof

By Lemmas 2.1 and 2.2, every maximal subgroup of P is c-normal or Φ -supplemented in E . Obviously, ( ∣ E ∣ , p − 1 ) = 1 . By Theorem 3.1, E is p-nilpotent. Let T be the normal p-complement of E , then T is normal in G . Suppose that T ≠ 1 . Then, by Lemmas 2.1 and 2.2, the factor group G / T and its normal subgroup E / T satisfy the hypotheses. Thus, by induction, we have that G / T is p-nilpotent. It follows that G is p-nilpotent, as expected. Now, we suppose that T = 1 . Then, P = E . Let K / P be the normal p-complement of G / P . Then, K is normal in G and G / K is p-group. It is easy to see that K satisfies the hypotheses of Theorem 3.1. Hence, K is p-nilpotent. Let S be the normal p-complement of K . Because G / K is p-group, S is the normal p-complement of G , which implies that G is p-nilpotent.□

Theorem 3.2

Let P be a Sylow p-subgroup of G, where p is a prime divisor of ∣ G ∣ and ( ∣ G ∣ , p − 1 ) = 1 . Suppose that G is A 4 -free and every 2-maximal subgroup of P is c-normal or Φ -supplemented in G. Then, G is p-nilpotent.

Proof

Suppose that the assertion is not true, and let G be a counterexample with minimal order. By Lemma 2.4, p 3 ∣ ∣ G ∣ . We proceed via the following steps.

(1) G contains a unique minimal normal subgroup N with G / N p-nilpotent.

Let N be a minimal normal subgroup of G . Since P is a Sylow p-subgroup of G , P N / N is a Sylow p-subgroup of G ∕ N . Let M ∕ N be a 2-maximal subgroup of P N ∕ N and set H = M ∩ P . Then, M = M ∩ P N = ( M ∩ P ) N = H N and H ≤ P . Therefore,

∣ P : H ∣ = ∣ P : M ∩ P ∣ = ∣ P ∣ ∣ M ∣ ∣ P ∣ ∕ ∣ M P ∣ = ∣ M P ∣ ∣ M ∣ = ∣ N P ∕ N ∣ ∣ M ∕ N ∣ = p 2 ,

that is, H is a 2-maximal subgroup of P . By the hypotheses, there exists a normal subgroup T of G such that H ∩ T ≤ H G or H ∩ T ≤ Φ ( H ) . Since N is a minimal normal subgroup of G , we have that N ∩ T = 1 or N . Suppose first that N ∩ T = 1 . Then, ∣ N T ∣ = ∣ N ∣ ∣ T ∣ and ∣ N T ∣ ∣ ∣ G ∣ . Because, ∣ H T ∕ T ∣ = ∣ H ∕ H ∩ T ∣ is a power of p, N is an abelian subgroup of G and N < P . Thus, M = H is a 2-maximal subgroup of P and M is c-normal or Φ -supplemented in G . It follows that M ∕ N is c-normal or Φ -supplemented in G ∕ N by Lemmas 2.1 and 2.2. Now, we assume that N ≤ T . Since G = H T and T is normal in G , G ∕ N = H T ∕ N = ( M ∕ N ) ( T ∕ N ) and T ∕ N is normal in G ∕ N . If H ∩ T ≤ H G , then ( M ∕ N ) ∩ ( T ∕ N ) = ( H N ∩ T ) ∕ N = ( H ∩ T ) N ∕ N ≤ H G N ∕ N ≤ ( H N ∕ N ) ( G ∕ N ) = ( M ∕ N ) ( G ∕ N ) . If H ∩ T ≤ Φ ( H ) , then ( M ∕ N ) ∩ ( T ∕ N ) = ( H N ∩ T ) ∕ N = ( H ∩ T ) N ∕ N ≤ Φ ( H ) N ∕ N ≤ Φ ( H N ∕ N ) = Φ ( M ∕ N ) . Therefore, M ∕ N is c-normal or Φ -supplemented in G ∕ N . Obviously, ( ∣ G ∕ N ∣ , p − 1 ) = 1 and G ∕ N is A 4 -free. Hence, G ∕ N satisfies the hypotheses. By the choice of G , G ∕ N is p-nilpotent. Because all p-nilpotent groups form a saturated formation, N is unique in G .

(2) O p ( G ) ≠ 1 .

Suppose that O p ( G ) = 1 . Let H be a 2-maximal subgroup of P and H is normal in P . Then, Φ ( H ) ≤ Φ ( P ) . By the hypotheses, there exists a normal subgroup T of G such that G = H T and H ∩ T ≤ H G or H ∩ T ≤ Φ ( H ) . Since O p ( G ) = 1 and H is a p-subgroup, H G = 1 . Therefore, H ∩ T ≤ Φ ( H ) . Since Φ ( H ) < H , T < G . Because G ∕ T = H T ∕ T ≅ H ∕ H ∩ T is p-subgroup, G ∕ T is p-subgroup. We can take a maximal normal subgroup M ∕ T of G ∕ T such that ∣ G : M ∣ = p . Set K = M ∩ P and L is a maximal subgroup of K . Since G = H T and T ≤ M , G = H M . Because p = ∣ K : L ∣ = ∣ ( M ∩ P ) : L ∣ = ∣ M ∣ ∣ P ∣ ∣ M P ∣ ∣ L ∣ = ∣ M ∣ ∣ P ∣ ∣ G ∣ ∣ L ∣ = ∣ P ∣ p ∣ L ∣ , L is a 2-maximal subgroup of P . By the hypotheses, L is c-normal or Φ -supplemented in G . It follows that L is c-normal or Φ -supplemented in M by Lemmas 2.1 and 2.2. Since ( ∣ G ∣ , p − 1 ) = 1 and M < G , ( ∣ M ∣ , p − 1 ) = 1 . The foregoing arguments show that M satisfies the hypotheses. By the choice of G , M is p-nilpotent. Let S be the normal p-complement of M . Because G ∕ M is p-group, S is the normal p-complement of G . This contradiction shows that O p ( G ) ≠ 1 .

(3) The final contradiction.

By (1) and (2), N is the unique minimal normal subgroup of G and N ≤ O p ( G ) . Let H be a 2-maximal subgroup of P . By the hypotheses, there exists a normal subgroup T of G such that G = H T and H ∩ T ≤ H G or H ∩ T ≤ Φ ( H ) . If T < G , discussing as in Step (2), one can prove that G is p-nilpotent, a contradiction. Thus, T = G . It follows that H = H ∩ T = H G is normal in G . By (1) and (2), N is the unique minimal normal subgroup of G , N ≤ O p ( G ) , and there exists a maximal subgroup M of G such that G = [ N ] M . Let K be a Sylow p-subgroup of M such that P = [ N ] K . Let A be a maximal subgroup of N and A be normal in P . Let B be a maximal subgroup of K . Thus, A B ≤ P and A B is a 2-maximal subgroup of P . The choice of N shows that N ≤ A B since A B is normal in G by previous arguments, a final contradiction.□

Corollary 3.6

Let p be a prime dividing the order of G with ( ∣ G ∣ , p − 1 ) = 1 and E be a normal subgroup of G such that G/E is p-nilpotent. Suppose that P is a Sylow p-subgroup of E and every 2-maximal subgroup of P is c-normal or Φ -supplemented in G. If G is A 4 -free, then G is p-nilpotent.

Proof

By arguments similar to those used in the proof of Corollary 3.5, one can prove this result.

Theorem 3.3

Assume that P is a Sylow p-subgroup of G, where p is a prime divisor of ∣ G ∣ . Suppose that N G ( P ) is p-nilpotent and every maximal subgroup of P is c-normal or Φ -supplemented in G. Then, G is p-nilpotent.

Proof

If p = 2 , then by Theorem 3.1, G is p-nilpotent. Now we prove the theorem for the case of odd prime p. Suppose that the statement is not true, and let G be a counterexample of minimal order. If p 3 ∤ ∣ G ∣ , then P is abelian. Let K be the normal p-complement of N G ( P ) , then N G ( P ) = P × K . Thus, [ P , H ] = 1 . It follows that C G ( P ) = P × K = N G ( P ) . By the famous theorem of Burnside, G is p-nilpotent. Thus, p 3 ∣ ∣ G ∣ . We proceed via the following steps.

(1) O p ′ ( G ) = 1 .

If O p ′ ( G ) ≠ 1 , by Lemmas 2.1 and 2.2, G ∕ O p ′ ( G ) satisfies the hypotheses. The choice of G yields that G ∕ O p ′ ( G ) is p-nilpotent. Consequently, G is p-nilpotent, a contradiction. Hence, O p ′ ( G ) = 1 .

(2) If M is a proper subgroup of G with P ≤ M , then M is p-nilpotent.

Since N M ( P ) = N G ( P ) ∩ M and N G ( P ) is p-nilpotent, N M ( P ) is p-nilpotent. By Lemmas 2.1 and 2.2, M satisfies the hypotheses. The choice of G yields that M is p-nilpotent.

(3) G is not a non-abelian simple group and G has unique minimal normal subgroup N . Moreover, G ∕ N is p-nilpotent and Φ ( G ) = 1 .

Let H be a maximal subgroup of P . By the hypotheses, there exists a normal subgroup T of G such that G = H T and H ∩ T ≤ H G or H ∩ T ≤ Φ ( H ) . If T = G , then H ∩ T = H G is normal in G . Otherwise, H = 1 and ∣ P ∣ = p , which contradicts the fact that p 3 ∣ ∣ G ∣ by previous argument. If T ≠ G , then T is a proper subgroup of G and T ⊴ G . Therefore, G is not a non-abelian simple group. By arguments similar to those used in the proof of Theorem 3.1, one can see that the remaining assertions hold.

(4) G = P Q is solvable, where Q is a Sylow q -subgroup of G with q ≠ p .

Since G is not p-nilpotent, by [12, Corollary], there exists a characteristic subgroup L of P such that N G ( L ) is not p-nilpotent. By (2), N G ( L ) = G . This leads to N ≤ L . By (3), G is p-solvable. Then, for any q ∈ π ( G ) and q ≠ p , there exists a Sylow q -subgroup Q of G such that K = P Q is a subgroup of G . If K ≠ G , then by (2), K is p-nilpotent. By [13, Theorem 9.3.1], Q ≤ C G ( O p ( G ) ) ≤ O p ( G ) , a contradiction. Thus, G = K = P Q is solvable.

(5) The final contradiction.

By (1) and (2), N is the unique minimal normal subgroup of G and N ≤ O p ( G ) . By Step (3), there exists a maximal subgroup M of G such that G = M N and M ∩ N = 1 . Since N is an elementary abelian p-group, N ≤ C G ( N ) and C G ( N ) ∩ M ⊴ G . By the uniqueness of N , we have C G ( N ) ∩ M = 1 and N = C G ( N ) . But N ≤ O p ( G ) ≤ F ( G ) ≤ C G ( N ) , hence N = O p ( G ) = C G ( N ) . If P ∩ M = P , then N ≤ P ≤ M , a contradiction. Thus, we take a maximal subgroup H of P such that P ∩ M ≤ H . If P ∩ M = 1 , then P = N . It follows that N G ( P ) = G is p-nilpotent, a contradiction. Therefore, P ∩ M ≠ 1 . By the hypotheses, there exists a normal subgroup T of G such that H ∩ T ≤ H G or H ∩ T ≤ Φ ( H ) . By the uniqueness of N , N ≤ T . We assert that ∣ N ∣ = p .

If H ∩ T ≤ H G , then H ∩ N = H ∩ T ∩ N ≤ H G ∩ N ≤ H ∩ N . Consequently, we have that H ∩ N = H G ∩ N is normal in G , and therefore, H ∩ N = N or H ∩ N = 1 . Assume that H ∩ N = N . Then, N ≤ H . Since P = P ∩ N M = N ( P ∩ M ) and P ∩ M ≤ H , P = H . This contradiction shows that H ∩ N = 1 . Since P = P ∩ N M = N ( P ∩ M ) = N H and ∣ N : H ∩ N ∣ = ∣ N H : H ∣ = ∣ P : H ∣ = p , ∣ N ∣ = p .

If H ∩ T ≤ Φ ( H ) , then H = H ∩ P = H ∩ N ( P ∩ M ) ≤ ( H ∩ N ) ( P ∩ M ) ≤ ( H ∩ T ) H ≤ Φ ( H ) H = H . Thus, P ∩ M = H and ∣ N ∣ = p .

Since ∣ N ∣ = p , Aut ( N ) is cyclic of order p − 1 . If q > p , then H Q is p-nilpotent, and thus Q ≤ C G ( N ) = N , a contradiction. On the other hand, if q < p , then M ≅ G ∕ N = N G ( N ) ∕ C G ( N ) is isomorphic to a subgroup of Aut ( N ) because N = C G ( N ) . Hence, M and, in particular, Q are cyclic. Since Q is a cyclic group and q < p , we know that G is q -nilpotent. It follows that P is normal in G . This implies that N G ( P ) = G is p-nilpotent, a final contradiction.□

Corollary 3.7

Let E be a normal subgroup of G such that G ∕ E is p-nilpotent, where p is a prime divisor of ∣ G ∣ . Suppose that P is a Sylow p-subgroup of E , N G ( P ) is p-nilpotent, and every maximal subgroup of P is c-normal or Φ -supplemented in G . Then, G is p-nilpotent.

Proof

Since N E ( P ) = E ∩ N G ( P ) and N G ( P ) is p-nilpotent, N E ( P ) is p-nilpotent. By Lemmas 2.1 and 2.2, every maximal subgroup of P is c-normal or Φ -supplemented in E . By Theorem 3.3, E is p-nilpotent. Let T be the normal p-complement of E . Then, T is normal in G . By using the arguments used in the proof of Corollary 3.5, we may assume that T = 1 and E = P is a p-group. In this case, by our hypotheses, N G ( P ) = G is p-nilpotent.□

Acknowledgements

The authors wish to thank the referees for their useful comments.

  1. Funding information: The project was partially supported by the Training Program for Young Cadre Teachers of Higher Education Institutions in Henan Province (2020GGJS287), the Scientific Research Foundation for Advanced Talents of Suqian University (2022XRC069), the Science and Technology Research Program of Chongqing Municipal Education Commission (KJZD-K202001303) and sponsored by Natural Science Foundation of Chongqing, China (cstc2021jcyj-msxmX0511).

  2. Conflict of interest: The authors state that there is no conflict of interest.

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Received: 2021-11-12
Revised: 2022-11-27
Accepted: 2022-12-03
Published Online: 2022-12-31

© 2022 the author(s), published by De Gruyter

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  9. On certain functional equation in prime rings
  10. General complex Lp projection bodies and complex Lp mixed projection bodies
  11. Some results on the total proper k-connection number
  12. The stability with general decay rate of hybrid stochastic fractional differential equations driven by Lévy noise with impulsive effects
  13. Well posedness of magnetohydrodynamic equations in 3D mixed-norm Lebesgue space
  14. Strong convergence of a self-adaptive inertial Tseng's extragradient method for pseudomonotone variational inequalities and fixed point problems
  15. Generic uniqueness of saddle point for two-person zero-sum differential games
  16. Relational representations of algebraic lattices and their applications
  17. Explicit construction of mock modular forms from weakly holomorphic Hecke eigenforms
  18. The equivalent condition of G-asymptotic tracking property and G-Lipschitz tracking property
  19. Arithmetic convolution sums derived from eta quotients related to divisors of 6
  20. Dynamical behaviors of a k-order fuzzy difference equation
  21. The transfer ideal under the action of orthogonal group in modular case
  22. The multinomial convolution sum of a generalized divisor function
  23. Extensions of Gronwall-Bellman type integral inequalities with two independent variables
  24. Unicity of meromorphic functions concerning differences and small functions
  25. Solutions to problems about potentially Ks,t-bigraphic pair
  26. Monotonicity of solutions for fractional p-equations with a gradient term
  27. Data smoothing with applications to edge detection
  28. An ℋ-tensor-based criteria for testing the positive definiteness of multivariate homogeneous forms
  29. Characterizations of *-antiderivable mappings on operator algebras
  30. Initial-boundary value problem of fifth-order Korteweg-de Vries equation posed on half line with nonlinear boundary values
  31. On a more accurate half-discrete Hilbert-type inequality involving hyperbolic functions
  32. On split twisted inner derivation triple systems with no restrictions on their 0-root spaces
  33. Geometry of conformal η-Ricci solitons and conformal η-Ricci almost solitons on paracontact geometry
  34. Bifurcation and chaos in a discrete predator-prey system of Leslie type with Michaelis-Menten prey harvesting
  35. A posteriori error estimates of characteristic mixed finite elements for convection-diffusion control problems
  36. Dynamical analysis of a Lotka Volterra commensalism model with additive Allee effect
  37. An efficient finite element method based on dimension reduction scheme for a fourth-order Steklov eigenvalue problem
  38. Connectivity with respect to α-discrete closure operators
  39. Khasminskii-type theorem for a class of stochastic functional differential equations
  40. On some new Hermite-Hadamard and Ostrowski type inequalities for s-convex functions in (p, q)-calculus with applications
  41. New properties for the Ramanujan R-function
  42. Shooting method in the application of boundary value problems for differential equations with sign-changing weight function
  43. Ground state solution for some new Kirchhoff-type equations with Hartree-type nonlinearities and critical or supercritical growth
  44. Existence and uniqueness of solutions for the stochastic Volterra-Levin equation with variable delays
  45. Ambrosetti-Prodi-type results for a class of difference equations with nonlinearities indefinite in sign
  46. Research of cooperation strategy of government-enterprise digital transformation based on differential game
  47. Malmquist-type theorems on some complex differential-difference equations
  48. Disjoint diskcyclicity of weighted shifts
  49. Construction of special soliton solutions to the stochastic Riccati equation
  50. Remarks on the generalized interpolative contractions and some fixed-point theorems with application
  51. Analysis of a deteriorating system with delayed repair and unreliable repair equipment
  52. On the critical fractional Schrödinger-Kirchhoff-Poisson equations with electromagnetic fields
  53. The exact solutions of generalized Davey-Stewartson equations with arbitrary power nonlinearities using the dynamical system and the first integral methods
  54. Regularity of models associated with Markov jump processes
  55. Multiplicity solutions for a class of p-Laplacian fractional differential equations via variational methods
  56. Minimal period problem for second-order Hamiltonian systems with asymptotically linear nonlinearities
  57. Convergence rate of the modified Levenberg-Marquardt method under Hölderian local error bound
  58. Non-binary quantum codes from constacyclic codes over 𝔽q[u1, u2,…,uk]/⟨ui3 = ui, uiuj = ujui⟩
  59. On the general position number of two classes of graphs
  60. A posteriori regularization method for the two-dimensional inverse heat conduction problem
  61. Orbital stability and Zhukovskiǐ quasi-stability in impulsive dynamical systems
  62. Approximations related to the complete p-elliptic integrals
  63. A note on commutators of strongly singular Calderón-Zygmund operators
  64. Generalized Munn rings
  65. Double domination in maximal outerplanar graphs
  66. Existence and uniqueness of solutions to the norm minimum problem on digraphs
  67. On the p-integrable trajectories of the nonlinear control system described by the Urysohn-type integral equation
  68. Robust estimation for varying coefficient partially functional linear regression models based on exponential squared loss function
  69. Hessian equations of Krylov type on compact Hermitian manifolds
  70. Class fields generated by coordinates of elliptic curves
  71. The lattice of (2, 1)-congruences on a left restriction semigroup
  72. A numerical solution of problem for essentially loaded differential equations with an integro-multipoint condition
  73. On stochastic accelerated gradient with convergence rate
  74. Displacement structure of the DMP inverse
  75. Dependence of eigenvalues of Sturm-Liouville problems on time scales with eigenparameter-dependent boundary conditions
  76. Existence of positive solutions of discrete third-order three-point BVP with sign-changing Green's function
  77. Some new fixed point theorems for nonexpansive-type mappings in geodesic spaces
  78. Generalized 4-connectivity of hierarchical star networks
  79. Spectra and reticulation of semihoops
  80. Stein-Weiss inequality for local mixed radial-angular Morrey spaces
  81. Eigenvalues of transition weight matrix for a family of weighted networks
  82. A modified Tikhonov regularization for unknown source in space fractional diffusion equation
  83. Modular forms of half-integral weight on Γ0(4) with few nonvanishing coefficients modulo ℓ
  84. Some estimates for commutators of bilinear pseudo-differential operators
  85. Extension of isometries in real Hilbert spaces
  86. Existence of positive periodic solutions for first-order nonlinear differential equations with multiple time-varying delays
  87. B-Fredholm elements in primitive C*-algebras
  88. Unique solvability for an inverse problem of a nonlinear parabolic PDE with nonlocal integral overdetermination condition
  89. An algebraic semigroup method for discovering maximal frequent itemsets
  90. Class-preserving Coleman automorphisms of some classes of finite groups
  91. Exponential stability of traveling waves for a nonlocal dispersal SIR model with delay
  92. Existence and multiplicity of solutions for second-order Dirichlet problems with nonlinear impulses
  93. The transitivity of primary conjugacy in regular ω-semigroups
  94. Stability estimation of some Markov controlled processes
  95. On nonnil-coherent modules and nonnil-Noetherian modules
  96. N-Tuples of weighted noncommutative Orlicz space and some geometrical properties
  97. The dimension-free estimate for the truncated maximal operator
  98. A human error risk priority number calculation methodology using fuzzy and TOPSIS grey
  99. Compact mappings and s-mappings at subsets
  100. The structural properties of the Gompertz-two-parameter-Lindley distribution and associated inference
  101. A monotone iteration for a nonlinear Euler-Bernoulli beam equation with indefinite weight and Neumann boundary conditions
  102. Delta waves of the isentropic relativistic Euler system coupled with an advection equation for Chaplygin gas
  103. Multiplicity and minimality of periodic solutions to fourth-order super-quadratic difference systems
  104. On the reciprocal sum of the fourth power of Fibonacci numbers
  105. Averaging principle for two-time-scale stochastic differential equations with correlated noise
  106. Phragmén-Lindelöf alternative results and structural stability for Brinkman fluid in porous media in a semi-infinite cylinder
  107. Study on r-truncated degenerate Stirling numbers of the second kind
  108. On 7-valent symmetric graphs of order 2pq and 11-valent symmetric graphs of order 4pq
  109. Some new characterizations of finite p-nilpotent groups
  110. A Billingsley type theorem for Bowen topological entropy of nonautonomous dynamical systems
  111. F4 and PSp (8, ℂ)-Higgs pairs understood as fixed points of the moduli space of E6-Higgs bundles over a compact Riemann surface
  112. On modules related to McCoy modules
  113. On generalized extragradient implicit method for systems of variational inequalities with constraints of variational inclusion and fixed point problems
  114. Solvability for a nonlocal dispersal model governed by time and space integrals
  115. Finite groups whose maximal subgroups of even order are MSN-groups
  116. Symmetric results of a Hénon-type elliptic system with coupled linear part
  117. On the connection between Sp-almost periodic functions defined on time scales and ℝ
  118. On a class of Harada rings
  119. On regular subgroup functors of finite groups
  120. Fast iterative solutions of Riccati and Lyapunov equations
  121. Weak measure expansivity of C2 dynamics
  122. Admissible congruences on type B semigroups
  123. Generalized fractional Hermite-Hadamard type inclusions for co-ordinated convex interval-valued functions
  124. Inverse eigenvalue problems for rank one perturbations of the Sturm-Liouville operator
  125. Data transmission mechanism of vehicle networking based on fuzzy comprehensive evaluation
  126. Dual uniformities in function spaces over uniform continuity
  127. Review Article
  128. On Hahn-Banach theorem and some of its applications
  129. Rapid Communication
  130. Discussion of foundation of mathematics and quantum theory
  131. Special Issue on Boundary Value Problems and their Applications on Biosciences and Engineering (Part II)
  132. A study of minimax shrinkage estimators dominating the James-Stein estimator under the balanced loss function
  133. Representations by degenerate Daehee polynomials
  134. Multilevel MC method for weak approximation of stochastic differential equation with the exact coupling scheme
  135. Multiple periodic solutions for discrete boundary value problem involving the mean curvature operator
  136. Special Issue on Evolution Equations, Theory and Applications (Part II)
  137. Coupled measure of noncompactness and functional integral equations
  138. Existence results for neutral evolution equations with nonlocal conditions and delay via fractional operator
  139. Global weak solution of 3D-NSE with exponential damping
  140. Special Issue on Fractional Problems with Variable-Order or Variable Exponents (Part I)
  141. Ground state solutions of nonlinear Schrödinger equations involving the fractional p-Laplacian and potential wells
  142. A class of p1(x, ⋅) & p2(x, ⋅)-fractional Kirchhoff-type problem with variable s(x, ⋅)-order and without the Ambrosetti-Rabinowitz condition in ℝN
  143. Jensen-type inequalities for m-convex functions
  144. Special Issue on Problems, Methods and Applications of Nonlinear Analysis (Part III)
  145. The influence of the noise on the exact solutions of a Kuramoto-Sivashinsky equation
  146. Basic inequalities for statistical submanifolds in Golden-like statistical manifolds
  147. Global existence and blow up of the solution for nonlinear Klein-Gordon equation with variable coefficient nonlinear source term
  148. Hopf bifurcation and Turing instability in a diffusive predator-prey model with hunting cooperation
  149. Efficient fixed-point iteration for generalized nonexpansive mappings and its stability in Banach spaces
https://doi.org/10.1515/math-2022-0540
Keywords for this article
c-normal subgroups; Φ-supplemented subgroups; 2-maximal subgroups; p-nilpotency
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Articles in the same Issue

  1. Regular Articles
  2. A random von Neumann theorem for uniformly distributed sequences of partitions
  3. Note on structural properties of graphs
  4. Mean-field formulation for mean-variance asset-liability management with cash flow under an uncertain exit time
  5. The family of random attractors for nonautonomous stochastic higher-order Kirchhoff equations with variable coefficients
  6. The intersection graph of graded submodules of a graded module
  7. Isoperimetric and Brunn-Minkowski inequalities for the (p, q)-mixed geominimal surface areas
  8. On second-order fuzzy discrete population model
  9. On certain functional equation in prime rings
  10. General complex Lp projection bodies and complex Lp mixed projection bodies
  11. Some results on the total proper k-connection number
  12. The stability with general decay rate of hybrid stochastic fractional differential equations driven by Lévy noise with impulsive effects
  13. Well posedness of magnetohydrodynamic equations in 3D mixed-norm Lebesgue space
  14. Strong convergence of a self-adaptive inertial Tseng's extragradient method for pseudomonotone variational inequalities and fixed point problems
  15. Generic uniqueness of saddle point for two-person zero-sum differential games
  16. Relational representations of algebraic lattices and their applications
  17. Explicit construction of mock modular forms from weakly holomorphic Hecke eigenforms
  18. The equivalent condition of G-asymptotic tracking property and G-Lipschitz tracking property
  19. Arithmetic convolution sums derived from eta quotients related to divisors of 6
  20. Dynamical behaviors of a k-order fuzzy difference equation
  21. The transfer ideal under the action of orthogonal group in modular case
  22. The multinomial convolution sum of a generalized divisor function
  23. Extensions of Gronwall-Bellman type integral inequalities with two independent variables
  24. Unicity of meromorphic functions concerning differences and small functions
  25. Solutions to problems about potentially Ks,t-bigraphic pair
  26. Monotonicity of solutions for fractional p-equations with a gradient term
  27. Data smoothing with applications to edge detection
  28. An ℋ-tensor-based criteria for testing the positive definiteness of multivariate homogeneous forms
  29. Characterizations of *-antiderivable mappings on operator algebras
  30. Initial-boundary value problem of fifth-order Korteweg-de Vries equation posed on half line with nonlinear boundary values
  31. On a more accurate half-discrete Hilbert-type inequality involving hyperbolic functions
  32. On split twisted inner derivation triple systems with no restrictions on their 0-root spaces
  33. Geometry of conformal η-Ricci solitons and conformal η-Ricci almost solitons on paracontact geometry
  34. Bifurcation and chaos in a discrete predator-prey system of Leslie type with Michaelis-Menten prey harvesting
  35. A posteriori error estimates of characteristic mixed finite elements for convection-diffusion control problems
  36. Dynamical analysis of a Lotka Volterra commensalism model with additive Allee effect
  37. An efficient finite element method based on dimension reduction scheme for a fourth-order Steklov eigenvalue problem
  38. Connectivity with respect to α-discrete closure operators
  39. Khasminskii-type theorem for a class of stochastic functional differential equations
  40. On some new Hermite-Hadamard and Ostrowski type inequalities for s-convex functions in (p, q)-calculus with applications
  41. New properties for the Ramanujan R-function
  42. Shooting method in the application of boundary value problems for differential equations with sign-changing weight function
  43. Ground state solution for some new Kirchhoff-type equations with Hartree-type nonlinearities and critical or supercritical growth
  44. Existence and uniqueness of solutions for the stochastic Volterra-Levin equation with variable delays
  45. Ambrosetti-Prodi-type results for a class of difference equations with nonlinearities indefinite in sign
  46. Research of cooperation strategy of government-enterprise digital transformation based on differential game
  47. Malmquist-type theorems on some complex differential-difference equations
  48. Disjoint diskcyclicity of weighted shifts
  49. Construction of special soliton solutions to the stochastic Riccati equation
  50. Remarks on the generalized interpolative contractions and some fixed-point theorems with application
  51. Analysis of a deteriorating system with delayed repair and unreliable repair equipment
  52. On the critical fractional Schrödinger-Kirchhoff-Poisson equations with electromagnetic fields
  53. The exact solutions of generalized Davey-Stewartson equations with arbitrary power nonlinearities using the dynamical system and the first integral methods
  54. Regularity of models associated with Markov jump processes
  55. Multiplicity solutions for a class of p-Laplacian fractional differential equations via variational methods
  56. Minimal period problem for second-order Hamiltonian systems with asymptotically linear nonlinearities
  57. Convergence rate of the modified Levenberg-Marquardt method under Hölderian local error bound
  58. Non-binary quantum codes from constacyclic codes over 𝔽q[u1, u2,…,uk]/⟨ui3 = ui, uiuj = ujui⟩
  59. On the general position number of two classes of graphs
  60. A posteriori regularization method for the two-dimensional inverse heat conduction problem
  61. Orbital stability and Zhukovskiǐ quasi-stability in impulsive dynamical systems
  62. Approximations related to the complete p-elliptic integrals
  63. A note on commutators of strongly singular Calderón-Zygmund operators
  64. Generalized Munn rings
  65. Double domination in maximal outerplanar graphs
  66. Existence and uniqueness of solutions to the norm minimum problem on digraphs
  67. On the p-integrable trajectories of the nonlinear control system described by the Urysohn-type integral equation
  68. Robust estimation for varying coefficient partially functional linear regression models based on exponential squared loss function
  69. Hessian equations of Krylov type on compact Hermitian manifolds
  70. Class fields generated by coordinates of elliptic curves
  71. The lattice of (2, 1)-congruences on a left restriction semigroup
  72. A numerical solution of problem for essentially loaded differential equations with an integro-multipoint condition
  73. On stochastic accelerated gradient with convergence rate
  74. Displacement structure of the DMP inverse
  75. Dependence of eigenvalues of Sturm-Liouville problems on time scales with eigenparameter-dependent boundary conditions
  76. Existence of positive solutions of discrete third-order three-point BVP with sign-changing Green's function
  77. Some new fixed point theorems for nonexpansive-type mappings in geodesic spaces
  78. Generalized 4-connectivity of hierarchical star networks
  79. Spectra and reticulation of semihoops
  80. Stein-Weiss inequality for local mixed radial-angular Morrey spaces
  81. Eigenvalues of transition weight matrix for a family of weighted networks
  82. A modified Tikhonov regularization for unknown source in space fractional diffusion equation
  83. Modular forms of half-integral weight on Γ0(4) with few nonvanishing coefficients modulo ℓ
  84. Some estimates for commutators of bilinear pseudo-differential operators
  85. Extension of isometries in real Hilbert spaces
  86. Existence of positive periodic solutions for first-order nonlinear differential equations with multiple time-varying delays
  87. B-Fredholm elements in primitive C*-algebras
  88. Unique solvability for an inverse problem of a nonlinear parabolic PDE with nonlocal integral overdetermination condition
  89. An algebraic semigroup method for discovering maximal frequent itemsets
  90. Class-preserving Coleman automorphisms of some classes of finite groups
  91. Exponential stability of traveling waves for a nonlocal dispersal SIR model with delay
  92. Existence and multiplicity of solutions for second-order Dirichlet problems with nonlinear impulses
  93. The transitivity of primary conjugacy in regular ω-semigroups
  94. Stability estimation of some Markov controlled processes
  95. On nonnil-coherent modules and nonnil-Noetherian modules
  96. N-Tuples of weighted noncommutative Orlicz space and some geometrical properties
  97. The dimension-free estimate for the truncated maximal operator
  98. A human error risk priority number calculation methodology using fuzzy and TOPSIS grey
  99. Compact mappings and s-mappings at subsets
  100. The structural properties of the Gompertz-two-parameter-Lindley distribution and associated inference
  101. A monotone iteration for a nonlinear Euler-Bernoulli beam equation with indefinite weight and Neumann boundary conditions
  102. Delta waves of the isentropic relativistic Euler system coupled with an advection equation for Chaplygin gas
  103. Multiplicity and minimality of periodic solutions to fourth-order super-quadratic difference systems
  104. On the reciprocal sum of the fourth power of Fibonacci numbers
  105. Averaging principle for two-time-scale stochastic differential equations with correlated noise
  106. Phragmén-Lindelöf alternative results and structural stability for Brinkman fluid in porous media in a semi-infinite cylinder
  107. Study on r-truncated degenerate Stirling numbers of the second kind
  108. On 7-valent symmetric graphs of order 2pq and 11-valent symmetric graphs of order 4pq
  109. Some new characterizations of finite p-nilpotent groups
  110. A Billingsley type theorem for Bowen topological entropy of nonautonomous dynamical systems
  111. F4 and PSp (8, ℂ)-Higgs pairs understood as fixed points of the moduli space of E6-Higgs bundles over a compact Riemann surface
  112. On modules related to McCoy modules
  113. On generalized extragradient implicit method for systems of variational inequalities with constraints of variational inclusion and fixed point problems
  114. Solvability for a nonlocal dispersal model governed by time and space integrals
  115. Finite groups whose maximal subgroups of even order are MSN-groups
  116. Symmetric results of a Hénon-type elliptic system with coupled linear part
  117. On the connection between Sp-almost periodic functions defined on time scales and ℝ
  118. On a class of Harada rings
  119. On regular subgroup functors of finite groups
  120. Fast iterative solutions of Riccati and Lyapunov equations
  121. Weak measure expansivity of C2 dynamics
  122. Admissible congruences on type B semigroups
  123. Generalized fractional Hermite-Hadamard type inclusions for co-ordinated convex interval-valued functions
  124. Inverse eigenvalue problems for rank one perturbations of the Sturm-Liouville operator
  125. Data transmission mechanism of vehicle networking based on fuzzy comprehensive evaluation
  126. Dual uniformities in function spaces over uniform continuity
  127. Review Article
  128. On Hahn-Banach theorem and some of its applications
  129. Rapid Communication
  130. Discussion of foundation of mathematics and quantum theory
  131. Special Issue on Boundary Value Problems and their Applications on Biosciences and Engineering (Part II)
  132. A study of minimax shrinkage estimators dominating the James-Stein estimator under the balanced loss function
  133. Representations by degenerate Daehee polynomials
  134. Multilevel MC method for weak approximation of stochastic differential equation with the exact coupling scheme
  135. Multiple periodic solutions for discrete boundary value problem involving the mean curvature operator
  136. Special Issue on Evolution Equations, Theory and Applications (Part II)
  137. Coupled measure of noncompactness and functional integral equations
  138. Existence results for neutral evolution equations with nonlocal conditions and delay via fractional operator
  139. Global weak solution of 3D-NSE with exponential damping
  140. Special Issue on Fractional Problems with Variable-Order or Variable Exponents (Part I)
  141. Ground state solutions of nonlinear Schrödinger equations involving the fractional p-Laplacian and potential wells
  142. A class of p1(x, ⋅) & p2(x, ⋅)-fractional Kirchhoff-type problem with variable s(x, ⋅)-order and without the Ambrosetti-Rabinowitz condition in ℝN
  143. Jensen-type inequalities for m-convex functions
  144. Special Issue on Problems, Methods and Applications of Nonlinear Analysis (Part III)
  145. The influence of the noise on the exact solutions of a Kuramoto-Sivashinsky equation
  146. Basic inequalities for statistical submanifolds in Golden-like statistical manifolds
  147. Global existence and blow up of the solution for nonlinear Klein-Gordon equation with variable coefficient nonlinear source term
  148. Hopf bifurcation and Turing instability in a diffusive predator-prey model with hunting cooperation
  149. Efficient fixed-point iteration for generalized nonexpansive mappings and its stability in Banach spaces
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