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On regular subgroup functors of finite groups

  • Baojun Li , Yan Wu and Lü Gong EMAIL logo
Published/Copyright: December 31, 2022
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Open Mathematics
From the journal Open Mathematics Volume 20 Issue 1

Abstract

A subgroup functor τ is said Φ -regular if for all primitive groups G , whenever H ∈ τ ( G ) is a p -subgroup and N is a minimal normal subgroup of G , then ∣ G : N G ( H ∩ N ) ∣ = p d for some integer d . In this article, we investigate groups in which some primary subgroups are τ -subgroups for a Φ -regular subgroup functor τ , and we obtain new criteria for the supersolubility or p -nilpotency of a group.

Keywords: finite group; subgroup functor; primary subgroup; 𝔉-hypercenter
MSC 2010: 20D10

1 Introduction

Groups in this article are all finite. The reader is referred to [1,2] for unexplained notations and terminologies. Let G be a group and p a prime. Then G p denotes a Sylow p -subgroup of G and π ( G ) is the set of all primes dividing ∣ G ∣ . Recall that τ is said a subgroup functor if 1 ∈ τ ( G ) is a subset of subgroups of G and θ ( τ ( G ) ) = τ ( θ ( G ) ) for every isomorphism θ : G ⟶ G ∗ . If H ∈ τ ( G ) , then H is said a τ -subgroup of G [2]. The sets of all Sylow subgroups, maximal subgroups, subnormal subgroups, permutable subgroups, and so on are some known subgroup functors. Investigating the influence of subgroup functor, the notions of regular and Φ -regular subgroup functors are proposed in [2], and some interesting results were obtained in [2,3].

Definition 1.1

[2] Let τ be a subgroup functor. Then τ is said:

  1. regular if for any group G , whenever H ∈ τ ( G ) is a p -subgroup and N is a minimal normal subgroup of G , then ∣ G : N G ( H ∩ N ) ∣ is a power of p .

  2. Φ -regular if for any primitive group G , whenever H ∈ τ ( G ) is a p -subgroup and N is a minimal normal subgroup of G , then ∣ G : N G ( H ∩ N ) ∣ is a power of p .

Let τ ( G ) be the set of all normal subgroups, permutable subgroups, or s -permutable subgroups. Then τ is regular. Recall that a primitive group is a group in which some maximal subgroups have trivial core. Clearly, if a subgroup functor is regular, then it must be Φ -regular, but the converse is not true in general. For example, let τ ( G ) = { H ∣ H ≤ Φ ( G ) } for all group G . Then τ ( G ) is Φ -regular, but it is not regular. Generally, if τ 1 is regular and τ ( G ) = { H K ∣ H ≤ Φ ( G ) , K ∈ τ 1 ( G ) and H K = K H } for all groups G , then τ is Φ -regular.

Let H / K be a chief factor of a group G and F a class of groups. Recall that H / K is said F -central if ( H / K ) ⋊ ( G / C G ( H / K ) ) ∈ F . The largest normal subgroup of G whose G -chief factors are all F -central is F -hypercenter and is denoted by Z F ( G ) . Furthermore, the largest normal subgroup of G , in which all non-Frattini G -chief factors are F -central in G , is called the F Φ -hypercenter (compared with [4]), and denoted by Z F Φ ( G ) . Let U be the formation of all supersoluble groups. Then Z U ( G ) is the largest normal subgroup of G in which all G -chief factors are cyclic, and Z U Φ ( G ) is the largest normal subgroup in which all non-Frattini G -chief factors are cyclic. By investigating on subgroups in which some primary subgroups (i.e., subgroups of prime-power order) are Φ -regular, we obtain the following result.

Theorem 1.2

Let τ be a Φ -regular subgroup functor and E ⊴ G . If for all p ∈ π ( E ) , there is a p-subgroup D ( p ) with ∣ Φ ( G ) ∩ O p ( E ) ∣ < ∣ D ( p ) ∣ < ∣ E p ∣ such that all subgroups of E with order ∣ D ( p ) ∣ are contained in τ ( G ) , then E ≤ Z U Φ ( G ) .

Fixing a prime p , we have

Theorem 1.3

Let τ be a Φ -regular subgroup functor and E ⊴ G . Let p ∈ π ( E ) and assume that N G ( E p ) is p-nilpotent. If there is a p-subgroup D with ∣ Φ ( G ) ∩ O p ( E ) ∣ < ∣ D ∣ < ∣ E p ∣ such that all subgroups of E with order ∣ D ∣ are contained in τ ( G ) , then E ≤ Z F Φ ( G ) , where F is the formation of all p-nilpotent groups.

2 Preliminary results

The following results are known, and we list them here as lemmas.

Lemma 2.1

[5, III, Lemma 3.3] Let G be a group, N ⊴ G and U ≤ G . If N ≤ Φ ( U ) , then N ≤ Φ ( G ) .

Lemma 2.2

[6, Lemma 1.8.16] Let N be a nilpotent normal subgroup of G. If N ∩ Φ ( G ) = 1 , then N is complemented in G.

Recall that the generalized Fitting subgroup of a group G is the largest normal quasinilpotent subgroup (see [7]). Let F p ∗ ( G ) / O p ′ ( G ) = F ∗ ( G / O p ′ ( G ) ) . A formation is a class of groups F that is closed under subdirect products and epimorphic images, and that a formation F is saturated when G ∈ F whenever G / Φ ( G ) ∈ F , where Φ ( G ) denotes the Frattini subgroup of G . We have the following lemma.

Lemma 2.3

Let E ⊴ G . Assume that all G-chief factors H / K are cyclic whenever H ≤ F p ∗ ( E ) and p ∣ ∣ H / K ∣ . Then all G-chief factors L / M are cyclic whenever L ≤ E and p ∣ ∣ L / M ∣ .

Proof

Let G act on E by conjugation. By choosing f ( p ) = A ( p − 1 ) , the class of all abelian groups with exponents dividing p − 1 , it follows directly from [8, Corollary 4.3] that the lemma holds.□

The following result is a corollary of Lemma 2.3 and can also be found in [9].

Lemma 2.4

[9, Lemma 2.17] Let E be a normal subgroup of G. If F ∗ ( E ) ≤ Z U ( G ) , then E ≤ Z U ( G ) .

As we know, Z U Φ ( G ) = Z U ( G ) does not hold in general (see Example 1.2 in [4]). But we have the following result.

Lemma 2.5

Let X be a normal subgroup of G. If X / Φ ( X ) ≤ Z U ( G / Φ ( X ) ) , then X ≤ Z U ( G ) .

Proof

If Φ ( X ) = 1 , the assertion is clear. Assume that Φ ( X ) ≠ 1 and let N ≤ Φ ( X ) be a minimal normal subgroup of G . Then the hypotheses hold on ( G / N , X / N ) , and so, X / N ≤ Z U ( G / N ) by induction on ∣ X ∣ . Thus, it is enough to show that N is cyclic. If X possesses a minimal normal subgroup L different to N , then N L / L ≤ Φ ( X / L ) , and hence, ( X / L ) / Φ ( X / L ) ≤ Z U ( ( G / L ) / Φ ( X / L ) ) . Thus, X / L ≤ Z U ( G / L ) by induction on ∣ X ∣ . It follows that N ≅ N L / L is cyclic. Assume that N is the unique minimal normal subgroup of G contained in X . Since X / Φ ( X ) ≤ Z U ( G / Φ ( X ) ) , X is supersoluble by [9, Lemma 2.16]. Let p be the largest prime divisor of ∣ X ∣ . Then X p ⊴ X and N ≤ X p . If Φ ( X p ) ≠ 1 then N ≤ Φ ( X p ) . Since X p / N ≤ X / N ≤ Z U ( G / N ) , it follows directly from [10, Lemma 2.8] that X p ≤ Z U ( G ) , and hence, N is cyclic. Assume that Φ ( X p ) = 1 . Then by Maschke’s theorem, N has an X -invariant complement in X p . This is nonsense for N ≤ Φ ( X ) . Hence, X ≤ Z U ( G ) , and the lemma holds.□

Lemma 2.6

Let E ⊴ G . If E ∩ Φ ( G ) = Φ ( E ) , then E ≤ Z U Φ ( G ) if and only if E ≤ Z U ( G ) .

Proof

The “if” part is clear, and we only prove the “only if” part. Assume E ∩ Φ ( G ) = Φ ( E ) ≠ 1 and let N ≤ Φ ( E ) be a minimal normal subgroup of G . Then E / N ∩ Φ ( G / N ) = Φ ( E / N ) and E / N ≤ Z U Φ ( G / N ) hold. By induction on ∣ E ∣ , we can obtain that E / N ≤ Z U ( G / N ) , and it follows that E ≤ Z U ( G ) by Lemma 2.5.

Assume E ∩ Φ ( G ) = Φ ( E ) = 1 . Let N be a minimal normal subgroup of G contained in E . Then N is cyclic since N ≤ E ≤ Z U Φ ( G ) and N ≰ Φ ( G ) . Clearly, E is soluble. By Lemma 2.2, we see that F ( E ) = F ∗ ( E ) is the product of some minimal normal subgroups of G . Hence, F ( E ) ≤ Z U ( G ) and so E ≤ Z U ( G ) by Lemma 2.4. This completes the proof of the lemma.□

By [4, Proposition 5.2] and [11, Theorem A], we have the following lemmas.

Lemma 2.7

[4] Let F be a saturated formation and E a normal subgroup with G / E ∈ F . If E ≤ Z F Φ ( G ) , then G ∈ F .

Lemma 2.8

[11] Let E be a normal subgroup of G. If F ( E ) ≤ Z U Φ ( G ) and E is soluble, then E ≤ Z U Φ ( G ) .

3 Proofs of Theorems 1.2 and 1.3

Lemma 3.1

Let all minimal normal subgroups of a group G be cyclic and Φ ( G ) = 1 . Then G is supersoluble.

Proof

Since Φ ( G ) = 1 and all minimal normal subgroups of G are cyclic, F ( G ) = Soc ( G ) is the product of all minimal normal subgroups of G . Assume that p ∣ ∣ F ( G ) ∣ and let P = O p ( G ) . By [1, A (10.6)], there exists a subgroup M of G such that G = F ( G ) ⋊ M . Let F ( G ) = N 1 × N 2 × ⋯ × N r . Since F ( G ) is abelian, F ( G ) ≤ C G ( F ( G ) ) and G / C G ( F ( G ) ) = M C G ( F ( G ) ) / C G ( F ( G ) ) ≅ M / C M ( F ( G ) ) = M / ⋂ i = 1 r C M ( N i ) is isomorphic to a subgroup of the group M / C M ( N 1 ) × M / C M ( N 2 ) × ⋯ × M / C M ( N r ) , that is abelian because all N i are cyclic. Let C = C M ( F ( G ) ) . We have that C is a normal subgroup of M . Assume that C ≠ 1 , and let R be a minimal normal subgroup of M contained in C . Then R is normal in F ( G ) M = G , and so R is a minimal normal subgroup of G . By hypothesis, R is cyclic, and so R ≤ F ( G ) ∩ M = 1 . This contradiction shows that C = 1 and, since M ≅ G / F ( G ) is abelian and F ( G ) is the product of cyclic minimal normal subgroups of G and G is supersoluble.□

Proof of Theorem 1.2

Assume that the theorem does not hold, and let G be a counterexample of minimal order. We prove the theorem via the following steps.

(1) All minimal normal subgroups of G are contained in E.

Let N be a minimal normal subgroup of G . Suppose that N ≰ E and H / N in E N / N is a subgroup of order ∣ D ( p ) ∣ for some prime p . Then ∣ H ∩ E ∣ = ∣ D ( p ) ∣ and hence H ∩ E ∈ τ ( G ) . Let θ : G ⟶ G / N be the natural epimorphism. Then H / N = ( H ∩ E ) N / N = θ ( H ∩ E ) ∈ θ ( τ ( G ) ) = τ ( θ ( G ) ) = τ ( G / N ) . ∣ Φ ( G / N ) ∩ O p ( E N / N ) ∣ < ∣ D ( p ) ∣ < ∣ E p N / N ∣ is clear. Thus, the hypotheses hold on G / N and hence E N / N ≤ Z U Φ ( G / N ) since ∣ G / N ∣ < ∣ G ∣ . It follows that E ≤ Z U Φ ( G ) , a contradiction, and hence (1) holds.

(2) Φ ( G ) = 1 .

Assume that Φ ( G ) ≠ 1 , and let N ≤ Φ ( G ) be a minimal normal subgroup of G and q ∣ ∣ N ∣ . In G ¯ = G / N , choose ∣ D ¯ ( p ) ∣ = ∣ D ( p ) ∣ if p ≠ q and ∣ D ¯ ( p ) ∣ = ∣ D ( p ) / N ∣ if p = q . Then ∣ Φ ( G ¯ ) ∩ O p ( E ¯ ) ∣ < ∣ D ¯ ( p ) ∣ < ∣ E ¯ p ∣ , and by a similar argument as (1), we can obtain that all subgroups in E ¯ of order ∣ D ¯ ( p ) ∣ are τ -subgroups. Thus, the hypotheses hold on G / N , and so, E / N ≤ Z U Φ ( G / N ) . Hence, E ≤ Z U Φ ( G ) . This contradicts the choice of G and (2) holds.

(3) If N is a minimal normal subgroup of G, then N is cyclic.

Since Φ ( G ) = 1 by (2), there is a maximal subgroup M such that G = N M . Then G / M G is primitive and N M G / M G is a minimal normal subgroup of G / M G . Let p be a prime divisor of N and H be a subgroup of order D ( p ) . Then, H ∈ τ ( G ) , and hence, H M G / M G ∈ τ ( G / M G ) by choosing θ to be the natural epimorphism of G to G / M G . Thus, ∣ G / M G : N G / M G ( ( H M G ∩ N M G ) / M G ) ∣ is a p -number. It follows that G / M G = ( G / M G ) p N G / M G ( ( H M G ∩ N M G ) / M G ) , where ( G / M G ) p is a Sylow p -subgroup of G / M G containing ( H M G ∩ N M G ) / M G . Then ( ( H M G ∩ N M G ) / M G ) G / M G ≤ ( G / M G ) p is a p -subgroup and so ( H M G ∩ N M G ) / M G ≤ O p ( G / M G ) . Since H ∩ N ≠ 1 , we see that ( N M G / M G ) ∩ O p ( G / M G ) ≠ 1 and so N ≅ N M G / M G is a p -group. Furthermore, we have N ∩ M = 1 and G = N ⋊ M . Thus, G p = N ⋊ M p . If ∣ D ( p ) ∣ < ∣ N ∣ , then let H 1 be a normal subgroup of G p contained in N with ∣ H 1 ∣ = ∣ D ( p ) ∣ and let H 2 = 1 ; if ∣ D ( p ) ∣ ≥ ∣ N ∣ , then let H 1 be a normal subgroup of G p and maximal in N , and let H 2 be a subgroup of M p of order ∣ D ( p ) ∣ / ∣ H 1 ∣ . Let H = H 1 H 2 . Then ∣ H ∣ = ∣ D ( p ) ∣ and H 1 = H ∩ N ⊴ G p . Moreover, it holds that H M G ∩ N M G = ( H M G ∩ N ) M G ≤ ( H M p ∩ N ) M G = ( H 1 M p ∩ N ) M G = H 1 M G . Since H 1 M G ≤ H M G ∩ N M G is clear, H M G ∩ N M G = H 1 M G ⊴ G p M G . Since H is Φ -regular in G , we see that ∣ G : N G ( H M G ∩ N M G ) ∣ = ∣ G / M G : N G / M G ( ( H M G ∩ N M G ) / M G ) ∣ is a p -number. Thus, H M G ∩ N M G ⊴ G = G p N G ( H M G ∩ N M G ) . It follows that H 1 = N ∩ H 1 M G ⊴ G . But N is minimal normal in G , so H 1 = 1 and N is cyclic.

(4) The final contradiction

By Lemma 3.1 and step (3), we see that G is supersoluble, and hence, E ≤ Z U Φ ( G ) , which contradicts the choice of G . This is the final contradiction and the theorem holds.□

Proof of Theorem 1.3

Assume that the theorem is not true, and let G be a counterexample of minimal order. Since the hypotheses still hold on G / Φ ( G ) and G / O p ′ ( E ) , Φ ( G ) = O p ′ ( E ) = 1 by the minimality of G . Let N be a minimal normal subgroup of G . If N ⊈ E , then E N / N ≤ Z F Φ ( G / N ) . It follows that E ≤ Z F Φ ( G ) , a contradiction. Thus, N ≤ E . Choose H to be a subgroup of order ∣ D ∣ , H ∩ N is maximal in N and is normal in a Sylow p -subgroup of G . Let M be a complement of N in G . Then ∣ G : N G ( H M G ∩ N M G ) ∣ = ∣ G / M G : N G / M G ( ( H M G ∩ N M G ) / M G ) ∣ is a p -number. It follows that H M G ∩ N M G ⊴ G = G p N G ( H M G ∩ N M G ) and H 1 = N ∩ H M G ⊴ G . But N is minimal normal in G , so H ∩ N = 1 and N is cyclic. By Lemma 3.1, E is supersoluble and E p ⊴ E . It follows that E p ⊴ G and G = N G ( E p ) is p -nilpotent by the hypotheses. Thus, E ≤ Z F Φ ( G ) , where F is the formation of all p -nilpotent groups.□

4 Some remarks, examples and applications

1. The following example shows that ∣ Φ ( G ) ∩ O p ( E ) ∣ < ∣ D ( p ) ∣ is necessary in Theorem 1.2.

Example 4.1

Let A = ⟨ a ∣ a p 2 = 1 ⟩ be a cyclic group of order p 2 and B = ⟨ b ∣ b q 2 = 1 ⟩ a cyclic group of order q 2 and q ∤ ( p − 1 ) . Assume that K = A 1 × A 2 × ⋯ × A q , where A i ≅ A and B acts on K with ( a 1 , a 2 , ⋯ , a q ) b = ( a q , a 1 , ⋯ , a q − 1 ) . Then G = K ⋊ B is non-supersoluble. Let τ ( G ) = { H ∣ H ≤ Φ ( G ) } . Then τ is Φ -regular, and all minimal subgroups of G are τ -subgroups since they are contained in Φ ( G ) . Let E = G . Then E ≤ Z U Φ ( G ) does not hold.

A similar example shows that ∣ Φ ( G ) ∩ O p ( E ) ∣ < ∣ D ∣ is necessary in Theorem 1.3.

Example 4.2

Let A = ⟨ a ∣ a p 2 = 1 ⟩ be a cyclic group of order p 2 , B = ⟨ b ∣ b q = 1 ⟩ a cyclic group of order q and M = A ≀ B be the regular wreath product of A and B . Let T = ⟨ x ∣ x p 2 = 1 ⟩ . Assume that R = M 1 × M 2 × ⋯ × M p , where M i ≅ M and T acts on R with ( a 1 , a 2 , ⋯ , a p ) x = ( a p , a 1 , ⋯ , a p − 1 ) . Then G = R ⋊ T is non- p -nilpotent. Let τ ( G ) = { H ∣ H ≤ Φ ( G ) } . Then τ is Φ -regular, and all minimal subgroups of order p of G are τ -subgroups since they are contained in Φ ( G ) . Let E = G . Then E ≤ Z F Φ ( G ) does not hold, where F is the formation of all p -nilpotent group.

2. In Theorem 1.2, E ≤ Z U ( G ) does not hold in general (by choosing E ¯ to be the U Φ -hypercenter of G ¯ = G / N in [4, Example 1.2]). But, by applying Lemma 2.6, we see that E ≤ Z U ( G ) if Φ ( G ) ∩ E = Φ ( E ) .

3. By Theorems 1.2 and 1.3, we have the following results.

Theorem 4.3

Let E be a normal subgroup of a group G and G/E supersoluble. Assume that τ is a Φ -regular subgroup functor. If for every prime divisor p of ∣ E ∣ , there is a p-subgroup D ( p ) with ∣ Φ ( G ) ∩ O p ( E ) ∣ < ∣ D ( p ) ∣ < ∣ E p ∣ such that all subgroups in E of order ∣ D ( p ) ∣ are contained in τ ( G ) , then G is supersoluble.

Proof

By Theorem 1.2, we see that E ≤ Z U Φ ( G ) , and it follows directly from Lemma 2.7 that G is supersoluble.□

Theorem 4.4

Let E be a soluble normal subgroup of a group G with G/E supersoluble. Assume that τ is a Φ -regular subgroup functor. If for every prime divisor p of F ∗ ( E ) , there is a p-subgroup D ( p ) with ∣ Φ ( G ) ∩ O p ( F ∗ ( E ) ) ∣ < ∣ D ( p ) ∣ < ∣ ( F ∗ ( E ) ) p ∣ such that all subgroups in E of order ∣ D ( p ) ∣ are contained in τ ( G ) , then G is supersoluble.

Proof

By Theorem 1.2 F ∗ ( E ) ≤ Z U Φ ( G ) , and hence, E ≤ Z U Φ ( G ) by Lemma 2.8. It follows from Lemma 2.7 that G is supersoluble.□

Theorem 4.5

Let E be a normal subgroup of a group G with p-nilpotent quotient and τ a Φ -regular subgroup functor. Let P be a Sylow p subgroup of E. If N G ( P ) is p-nilpotent, and there is a p-subgroup D with ∣ Φ ( G ) ∩ O p ( E ) ∣ < ∣ D ∣ < ∣ P ∣ such that all subgroups in E of order ∣ D ∣ are contained in τ ( G ) , then G is p-nilpotent.

Proof

By Theorem 1.3, E ≤ Z F Φ ( G ) , where F is the formation of all p -nilpotent groups, and hence, G is p -nilpotent by Lemma 2.7.□

Theorem 4.6

Let E be a normal subgroup of a group G with p-nilpotent quotient and τ a Φ -regular subgroup functor. Let P be a Sylow p subgroup of F p ∗ ( E ) . If N G ( P ) is p -nilpotent, and there is a p-subgroup D with ∣ Φ ( G ) ∩ O p ( E ) ∣ < ∣ D ∣ < ∣ P ∣ such that all subgroups in F p ∗ ( E ) of order ∣ D ∣ are contained in τ ( G ) , then G is p -nilpotent.

Proof

Let F be the formation of all p -nilpotent groups. By Theorem 1.3, F p ∗ ( E ) ≤ Z F Φ ( G ) is p -nilpotent and hence F p ∗ ( E ) = O p ′ ( E ) P . Since N G ( P ) is p -nilpotent, by the Frattini Argument, G = O p ′ ( E ) N G ( P ) is p -nilpotent and the theorem holds.□

4. It can be verified that in Theorems 1.3, 4.5 and 4.6 “ N G ( P ) is p -nilpotent” is not necessary if p is the minimal divisor of ∣ G ∣ .

5. Let τ be a group functor. If τ ( G ) is the set of all normal subgroups; permutable subgroups; s-permutable subgroups; c-normal subgroups; c-semipermtable subgroups (if G is soluble) or Z -permutable subgroups of G , then the subgroup functor τ is Φ -regular (compared with [2]). By choosing D ( p ) to be a maximal subgroup of a Sylow p -subgroup of E , our main results uniform a lot of known results (see some theorems in [13,14,15, 16,17,18, 19,20,21] and so on).

6. An s -permutably embedded subgroup (compared with [22], etc) is subgroup in which every Sylow subgroup is also a Sylow subgroup of some s -permutable subgroup. Let τ ( G ) = { H ∣ H is s-permutably embedded in G } . Then τ is not regular in the universe of all finite groups, but it is regular in the universe of all soluble groups (compared with [2, III, Example 1.9]). Thus, many works about s-permutably embedded subgroups can be generalized by regular subgroup functor.

7. We also observed that in the literature, there were some group functors that do not correspond to a regular subgroup functor. A subgroup H of G is said ℳ -supplemented in G if there exists a subgroup B of G such that G = H B and H 1 B < G for every maximal subgroup H 1 of H (compared with [23], etc.). Let τ ( G ) = { H ∣ H is ℳ -supplemented in G } . Since H ∈ τ ( G ) whenever H is a minimal supplement of a proper normal subgroup of G , τ ( G ) does not correspond to a regular subgroup functor in general. For example, Let G = S 5 be the symmetric group of degree 5, and H = ⟨ ( 1 2 3 4 ) ⟩ . Let B = A 5 . Then, G = H B and H 1 B < G for all proper subgroups H 1 of H . Thus, H is ℳ -supplemented in G . Clearly, ∣ G : N G ( H ∩ A 5 ∣ is not a 2-number.

Acknowledgements

The authors thank the referees for their useful comments.

  1. Funding information: The work was partially supported by the National Natural Science Foundation of China (11471055 and 11601245).

  2. Author contributions: All authors have accepted responsibility for the entire content of this manuscript and approved its submission.

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

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Received: 2022-05-18
Revised: 2022-12-13
Accepted: 2022-12-13
Published Online: 2022-12-31

© 2022 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. 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
https://doi.org/10.1515/math-2022-0549
Keywords for this article
finite group; subgroup functor; primary subgroup; 𝔉-hypercenter
Creative Commons
BY 4.0

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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