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BY 4.0 license Open Access Published by De Gruyter Open Access December 31, 2022

Some new characterizations of finite p-nilpotent groups

  • Fengyan Xie and Jinbao Li EMAIL logo
From the journal Open Mathematics

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.

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.

References

[1] O. Kegel, Produkte nilpotenter gruppen, Arch. Math. (Basel) 12 (1961), 90–93, https://doi.org/10.1007/bf01650529. Search in Google Scholar

[2] O. Kegel, On Huppert’s characterization of finite supersoluble groups, In: Proceedings of the International Conference on Theory Groups (Canberra, 1965), New York, 1967, pp. 209–215. Search in Google Scholar

[3] Y. Wang, C-normality of groups and its properties, J. Algebra 180 (1996), no. 3, 954–965, https://doi.org/10.1006/jabr.1996.0103. Search in Google Scholar

[4] Q. Yu, The Influence of the Properties of Local Finite Subgroups on the Structure of Finite Groups, Nanjing University Press, Nanjing, 2012. Search in Google Scholar

[5] D. Li and X. Guo, The influence of c-normality of subgroups on the structure of finite groups, J. Pure Appl. Algebra 150 (2000), no. 1, 53–60, https://doi.org/10.1016/s0022-4049(99)00042-0. Search in Google Scholar

[6] H. Liu and F. Xie, Φ-supplemented subgroups properties of local finite subgroups, J. Xinxiang University (Natural Science Edition) 33 (2016), 4–6. Search in Google Scholar

[7] M. Ramadan, M. Mohamed, and A. Heliel, On c-normality of certain subgroups of prime power order of finite groups, Arch. Math. (Basel) 85 (2005), 203–210, https://doi.org/10.1007/s00013-005-1330-1. Search in Google Scholar

[8] N. Su, Y. Wang, and Z. Xie, C-normal and hypercyclically embedded subgroups of finite groups, Bull. Malays. Math. Sci. Soc. 39 (2016), 1105–1113, https://doi.org/10.1007/s40840-015-0220-3. Search in Google Scholar

[9] J. Jaraden and A. Skiba, On c-normal subgroups of finite groups, Comm. Algebra 35 (2007), 3776–3788, https://doi.org/10.1080/00927870701353431. Search in Google Scholar

[10] B. Huppert, Endliche Gruppen, Springer-Verlag, Berlin-Heidelberg-New York, 1967. 10.1007/978-3-642-64981-3Search in Google Scholar

[11] B. Li and A. Skiba, New characterizations of finite supersoluble groups, Sci. China Ser. A-Math. 51 (2008), 827–841, https://doi.org/10.1007/s11425-007-0155-8. Search in Google Scholar

[12] J. Thompson, Normal p-complements for finite groups, J. Algebra 1 (1964), no. 1, 43–46, https://doi.org/10.1016/0021-8693(64)90006-7. Search in Google Scholar

[13] D. Robinson, A Course in the Theory of Groups, Springer-Verlag, Berlin-New York, 1967. Search in Google Scholar

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

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