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On sub-class sizes of mutually permutable products

  • Jinbao Li and Yong Yang EMAIL logo
Published/Copyright: May 3, 2021
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Open Mathematics
From the journal Open Mathematics Volume 19 Issue 1

Abstract

In this paper, we investigate the influence of sub-class sizes on a mutually permutable factorized group in which the sub-class sizes of some elements of its factors have certain quantitative properties. Some criteria for a group to be p -nilpotent or p -supersolvable are given.

Keywords: finite groups; sub-class sizes; mutually permutable products
MSC 2010: 20D10; 20D20; 20D40; 20E45

1 Introduction

All groups considered in this paper are finite.

We first fix some notation and terminology. Let G be a group. For an element x of G , we denote by x G = { x g ∣ g ∈ G } the conjugacy class of G containing x and denote by ∣ x G ∣ the class size in which x lies. If p is a prime, we say that x is a p -regular element or p ′ -element of G if p does not divide the order of x . We say that x is p -singular in G if its order is divisible by p , and a p -element of G is a p -singular element of prime power order. For a natural number n = p a ⋅ m with ( p , m ) = 1 , we call p a the p -part of n and denote it by n p . ∣ G ∣ denotes the order of G , and ∣ G ∣ p is the order of Sylow p -subgroups of G . Let H / K be a chief factor of G . If p divides ∣ H / K ∣ , then H / K is called a p -chief factor of G . In addition, π ( G ) denotes the set of all primes dividing the order of G . The other notation and terminology are standard.

The influence of certain arithmetical conditions of conjugacy class sizes on the structure of groups has been studied by many authors with lots of results available (see a nice survey of Camina and Camina [1] for the details). For example, Itô proved in [2] that for a group G and a prime p , ∣ x G ∣ is not divisible by p for every x ∈ G if and only if G has a central Sylow p -subgroup. If q is a prime different from p and p q does not divide ∣ x G ∣ for every x ∈ G , then G is either p -nilpotent or q -nilpotent. But the converse of this result is not true. Later on, Camina proved in [3] that ∣ x G ∣ p = 1 for every p ′ -element x ∈ G if and only if G = O p ( G ) × O p ′ ( G ) . This result was improved by Liu et al. in [4] for every p ′ -element x ∈ G of prime power order. It was also proved by Chillag and Herzog in [5] that G is supersolvable if ∣ x G ∣ is square-free for each x ∈ G . An improvement of this result was given in [4].

Recently, Qian and Yang in [6] generalized the usual concepts of “conjugacy class” and “conjugacy class size” and introduced the new concepts of “sub-class” and “sub-class size.” For a group G and an element x ∈ G , it is easy to observe that G has a unique minimal subnormal subgroup that contains x . We denote this subnormal subgroup by G x , which is uniquely associated with x . Instead of considering the class x G in G , one can just consider the class

x G ⋆ = { x g ∣ g ∈ G x } = x G x ,

which is called the sub-class of G containing x . ∣ x G ⋆ ∣ or ∣ x G x ∣ is called the sub-class size. By Remarks 1.8–1.10 in [6], we know that sub-classes and sub-class sizes have many interesting properties which are different from that of conjugacy classes and conjugacy class sizes. Therefore, it is natural to ask what happens for the structure of a group G if the restrictions on the usual conjugacy sizes of G as in the results mentioned in the last paragraph are replaced by the restrictions on the sub-class sizes. It was shown in [6] that some known results are still true and some new phenomena emerge as can be seen from the following two results (see [6, Theorem 1.4] and [6, Theorem 1.5], respectively).

Theorem 1.1

Let G be a group and p be a prime. Then, the following statements are equivalent.

  1. p does not divide ∣ x G ⋆ ∣ for every x of G .

  2. p does not divide ∣ x G ⋆ ∣ for every p ′ -element x ∈ G of prime power order.

  3. G is p -nilpotent.

Theorem 1.2

Suppose that G is a group and p , q are different primes. Then, p q does not divide ∣ x G ⋆ ∣ for each x of G if and only if G is either p -nilpotent or q -nilpotent.

Our purpose in this paper is to present some results on the structure of a mutually permutable factorized finite group in which the sub-class sizes of some elements of its factors have certain arithmetical properties. Recall that two subgroups A and B of a group G are said to be mutually permutable if A is permutable with all subgroups of B and B is permutable with all subgroups of A . Let G = G 1 G 2 ⋯ G r be the product of its pairwise permutable subgroups G 1 , G 2 , … , G r . Then, G is said to be the mutually permutable product of G 1 , G 2 , … , G r if G i and G j are mutually permutable subgroups of G for all i , j ∈ { 1 , 2 , … , r } . During the past three decades, the structure of mutually permutable products has been extensively developed according to the monograph by Ballester-Bolinches et al. [7]. Recently, Ballester-Bolinches et al. in [8] analyzed the structure of mutually permutable product G = G 1 G 2 ⋯ G r under the assumption that p or p 2 does not divide the conjugacy class sizes of some given elements in the factors G i for some fixed prime p . Theorems 1.1 and 1.3 in [8] together with the aforementioned Theorems 1.1 and 1.2 motivate us to further investigate the influence of sub-class sizes on p -nilpotence and p -supersolvability of a mutually permutable factorized group. Recall that a group G is called p -nilpotent if each p -chief factor H / K of G is central in G , that is H / K ⩽ Z ( G / K ) , or equivalently, if each Sylow p -subgroup of G has a normal p -complement in G . G is said to be p -supersolvable if each p -chief factor H / K has order p .

Our main results are as follows.

Theorem 1.3

Let G = G 1 G 2 ⋯ G r be the mutually permutable product of its subgroups G 1 , G 2 , … , G r and p be a prime. Then, p does not divide ∣ x G ⋆ ∣ for every p ′ -element x ∈ ⋃ i = 1 r G i of prime power order if and only if G is p -nilpotent.

Theorem 1.4

Let G = A B be the mutually permutable product of its p -solvable subgroups A and B and p be a prime. Suppose that for each p ′ -element x in A ∪ B , p 2 does not divide ∣ x G ⋆ ∣ . Then, G is p -supersolvable.

Theorem 1.5

Let G = A B be the mutually permutable product of its subgroups A and B and let p be a prime such that ( p − 1 , ∣ G ∣ ) = 1 . Suppose that p 2 does not divide ∣ x G ⋆ ∣ for each p ′ -element x ∈ A ∪ B of prime power order. Then, G is a solvable p -nilpotent group.

Theorem 1.3 can be viewed as an extension of Theorem 1.1 mentioned above by Qian and the second author. Theorem 1.4 is a sub-class size version of [8, Theorem 1.3] with the additional restriction that A and B are p -solvable. It was proved by Qian and Wang in [9] that for a group G and a prime p with ( p − 1 , ∣ G ∣ ) = 1 , if p 2 does not divide ∣ x G ∣ for every p ′ -element x ∈ G of prime power order, then G is a solvable p -nilpotent group and a Sylow p -subgroup of G / O p ( G ) is elementary abelian. Therefore, Theorem 1.5 can be partially regarded as a sub-class version of this result on mutually permutable products.

As an application of Theorems 1.4 and 1.5, we have the following corollary.

Corollary 1.6

Let G = A B be the mutually permutable product of its subgroups A and B . Suppose that for every p ∈ π ( G ) and each p ′ -element x ∈ A ∪ B , p 2 does not divide ∣ x G ⋆ ∣ . Then, G is supersolvable.

Proof

Let p be the smallest prime dividing the order of G . Then, the hypothesis and Theorem 1.5 imply that G is solvable. It follows from Theorem 1.4 that G is p -supersolvable for every prime p ∈ π ( G ) . Hence, G is supersolvable.□

For a group G and every element x ∈ G , we know from Remark 1.10 in [6] that ∣ x G ⋆ ∣ divides ∣ x G ∣ since G x is subnormal in G , and in many cases, ∣ x G ⋆ ∣ is much smaller that ∣ x G ∣ . As two immediate consequences of 1.6, we have:

Corollary 1.7

[8, Corollary 1.5] Let G = A B be the mutually permutable product of its subgroups A and B . Suppose that for every prime p ∈ π ( G ) and every p -regular element x ∈ A ∪ B , ∣ x G ∣ is not divisible by p 2 . Then, G is supersolvable.

Corollary 1.8

[4, Theorem 10] Let G = A B be a product of two subgroups A and B which are permutable in G . Suppose that for every prime p ∈ π ( G ) and every element x ∈ A ∪ B , ∣ x G ∣ is not divisible by p 2 . Then, G is supersolvable.

2 Preliminaries

In this section, we collect some results that will be used in our proof. We start by citing some basic properties about sub-classes and sub-class sizes that are contained in [6].

Lemma 2.1

Let G be a group and N be a subgroup of G . Let x be any element of G .

  1. If G is a simple group, then x G ⋆ = x G x = x G and therefore ∣ x G ⋆ ∣ = ∣ x G x ∣ = ∣ x G ∣ .

  2. If N is a subnormal subgroup of G such that x ∈ N , then N x = G x , x N ⋆ = x G ⋆ and ∣ x N ⋆ ∣ = ∣ x G ⋆ ∣ .

  3. Assume that N is normal in G and set G ¯ = G / N . Then, G ¯ x ¯ ⩽ G x ¯ , ∣ x ¯ G ¯ ⋆ ∣ ∣ ∣ x G ⋆ ∣ , and ∣ x ¯ G ¯ ⋆ ∣ ∣ ∣ x ¯ G ¯ ∣ ∣ ∣ x G ∣ .

Proof

(1) and (2) are obvious by the definition and (3) follows from [6, Lemma 2.1].□

Lemma 2.2

Let G = G 1 G 2 ⋯ G r be the product of the pairwise mutually permutable subgroups of G 1 , G 2 , … , G r . Then,

  1. If N is a normal subgroup of G , then G / N is the mutually permutable product of G 1 N / N , G 2 N / N , … , G r N / N .

  2. If each G i is p -solvable for a prime p , then G is p -solvable.

  3. If N is a non-abelian minimal normal subgroup of G , then there exists some j ∈ { 1 , 2 , … , r } such that N ⩽ G j .

Proof

Assertion (1) follows from [7, Theorem 4.1.11]. Assertion (2) follows from [7, Theorem 4.1.15]. Assertion (3) is [7, Theorem 4.3.8].□

For a group G , we let H G denote the largest normal subgroup of G contained in H , where H is a subgroup of G .

Lemma 2.3

If G = A B is a nontrivial group such that G is the product of the mutually permutable subgroups A and B , then A G B G is nontrivial.

Proof

See [7, Theorem 4.3.11].□

Lemma 2.4

Let G be a group and p be a prime. Then, p does not divide ∣ x G ∣ for every p ′ -element x ∈ G of prime power order if and only if G = O p ( G ) × O p ′ ( G ) .

Proof

See [4, Theorem 5].□

As usual, let F ( G ) denote the Fitting subgroup of a group G , which is generated by all nilpotent normal subgroups of G . Here, we denote by F p ( G ) the largest p -nilpotent normal subgroup of G , where p is a prime.

Lemma 2.5

Let G be a p -solvable group. Then, for given x ∈ G , ∣ x G ⋆ ∣ is not divisible by p if and only if x ∈ F p ( G ) .

Proof

This is [6, Lemma 3.1].□

Lemma 2.6

[10, Lemma 2.9] If P is a nilpotent normal subgroup of a group G and P ∩ Φ ( G ) = 1 , then P is the direct product of some minimal normal subgroups of G .

Lemma 2.7

Let p be a prime and A be a p ′ -group acting faithfully on an elementary abelian p -group N such that ∣ [ x , N ] ∣ = p for all nontrivial x ∈ A . Then, A is cyclic.

Proof

This is [8, Lemma 2.4].□

Lemma 2.8

Let S be a non-abelian simple group. Then, there exists an element x ∈ S of odd prime-power order such that 4 ∣ ∣ x S ∣ .

Proof

This is [9, Lemma 2.1].□

3 Proofs of main results

Proof of Theorem 1.3

The sufficiency part of Theorem 1.3 is clear in view of Theorem 1.1, and so we just need to consider the necessity part. Assume that it is false and let G be a counterexample of minimal order. By the hypotheses, we have that G = G 1 G 2 ⋯ G r is the mutually permutable product of its subgroups G 1 , G 2 , … , G r and p does not divide ∣ x G ⋆ ∣ for every p ′ -element x ∈ ⋃ i = 1 r G i of prime power order, where p is a prime. Let N be a minimal normal subgroup of G . Then, we can assume that N is non-trivial in G by Lemmas 2.2(3) and 2.4. By Lemma 2.2(1), G / N = ( G 1 N / N ) ( G 2 N / N ) ⋯ ( G r N / N ) is the mutually permutable product of G 1 N / N , G 2 N / N , … , G r N / N . In view of Lemma 2.1(3), we see that G / N satisfies the hypotheses and so G / N is p -nilpotent by the choice of G . If G has a minimal normal subgroup M different from N , then G / M is also p -nilpotent. It follows that G is p -nilpotent, since N ∩ M = 1 and so G = G / ( N ∩ M ) ≲ G / N × G / M , a contradiction. Hence, we have that N is the unique minimal normal subgroup of G .

We claim that N must be abelian. Otherwise, N ⩽ G j for some j ∈ { 1 , 2 , … , r } . Therefore, for every p ′ -element x ∈ N of prime power order, ∣ x N ⋆ ∣ = ∣ x G ⋆ ∣ is not divisible by p by the hypotheses. It follows from Theorem 1.1 that N is p -nilpotent and so N has a normal p -complement O p ′ ( N ) . This yields that N = O p ′ ( N ) by the minimality of N , which implies that G is p -nilpotent since G / N is p -nilpotent.

Hence, N is an abelian elementary p -group. Since G / N is p -nilpotent, we see that G is p -solvable. Since G is not p -nilpotent, we get that L = F p ( G ) < G . Let L p ′ be the normal p -complement of L . Then, by Lemma 2.5, we conclude that L p ′ contains every p ′ -element of prime power order in ⋃ i = 1 r G i and so L p ′ is nontrivial, a contradiction since L p ′ is normal in G , and N is the unique normal subgroup of G . Thus, the proof is complete.□

Proof of Theorem 1.4

Assume the result is false and let G be a counterexample of minimal order. By Lemma 2.2(2), we see that G is p -solvable.

Let N be an arbitrary nontrivial minimal normal subgroup of G . By Lemmas 2.1(3) and 2.2(1), we see that the quotient group G / N fulfills our conditions on G and so G / N is p -supersolvable by the minimality of G . Suppose that G has another minimal normal subgroup M with M ≠ N , then G / M is p -supersolvable by the foregoing argument. It follows that G is p -supersolvable since G = G / ( N ∩ M ) ≲ G / N × G / M . This contradiction shows that G has a unique minimal normal subgroup, say N , such that ∣ N ∣ ⩾ p 2 . We assert that Φ ( G ) = 1 . If not, then N ⩽ Φ ( G ) and so G / Φ ( G ) is p -supersolvable since G / N is p -supersolvable. It follows from [11, Satz 1] that G is p -supersolvable, a contradiction. Hence, we have Φ ( G ) = 1 . By Lemma 2.6, we see that F ( G ) is the direct product of some minimal normal subgroup of G . It follows that N = F ( G ) and so C G ( N ) = N by [12, Theorem 3.21].

Let L / N be a minimal normal subgroup of G / N = G / C G ( N ) . Since G / N is p -supersolvable, we have that L / N is either a p -group of order p or a p ′ -group. By [13, Ch. A, Lemma 13.6], we have that O p ( G / C G ( N ) ) = 1 , and therefore, G / N is a p ′ -chief factor of G . By Lemma 2.3, we may further assume that L / N ⩽ A N / N . We first suppose that L / N is not solvable. Then, L / N = L 1 / N × L 2 / N × ⋯ × L t / N is a direct product of isomorphic non-abelian simple groups L i / N for i ∈ { 1 , 2 , … , t } . It is clear that L i / N is perfect. Let x be a nontrivial p ′ -element in L 1 . Then, G x is contained in L 1 and N ⩽ N G ( G x ) by [12, Theorem 2.6]. Since G x is subnormal in G , we have that G x N / N = L 1 / N . Since L 1 / N is perfect and G x is normal in G x N , we see that N must be contained in G x and consequently L 1 / N = G x / N . Let L 1 = N ⋊ T , where T is a Hall p ′ -subgroup of L 1 . Then T ⩽ A and T acts faithfully on N . By the hypotheses, we have that p 2 does not divide ∣ x G ⋆ ∣ = ∣ x G x ∣ for every nontrivial x ∈ T and so ∣ N / C N ( x ) ∣ = p since N ⩽ G x . Now, invoking Lemma 2.7, we get that T is cyclic, a contradiction since L 1 / N is not solvable.

Now we assume that L / N is an elementary abelian p ′ -chief factor. Then, L = N ⋊ K for some elementary abelian q -group K with q a prime different from p . Without loss of generality, we assume that K ⩽ A by Lemma 2.3. Observe that K acts faithfully on N . Let x be any nontrivial element in K . Let J = N ⟨ x ⟩ . Then, J is subnormal in G and so G x ⩽ J . It follows that G x = G x ∩ N ⟨ x ⟩ = ( G x ∩ N ) ⟨ x ⟩ . Since N ⩽ N G ( G x ) by [12, Theorem 2.6], we have that G x ∩ N is normal in J . If G x ∩ N = 1 , then G x = ⟨ x ⟩ is normal in J , and therefore, x ∈ C G ( N ) = N , a contradiction. Hence, G x ∩ N ≠ 1 . By Maschke’s theorem (see [12, Corollary 10.17]), there exists a ⟨ x ⟩ -invariant subgroup U ⩽ N such that N = ( G x ∩ N ) × U . Since ⟨ x ⟩ acts completely reducibly on N , we can assume that U = U 1 × ⋯ × U r is a direct product of minimal normal subgroups U i of J for i = 1 , … , r . It is easy to see that U ∩ G x = 1 . By [12, Theorem 2.6] again, we have that U i ⩽ N J ( G x ) , and thereby, we get that U i ≤ C J ( G x ) . It follows that U ⩽ C J ( x ) . Since p 2 does not divide ∣ x G x ∣ by the hypotheses, we have that ∣ [ x , N ] ∣ = ∣ [ x , G x ∩ N ] ∣ = p . This holds for every nontrivial x ∈ K and so K is cyclic of order q by Lemma 2.7. Let K = ⟨ y ⟩ with y an element of order q . Then, L ′ = N and G y = L y is normalized by N by [12, Theorem 2.6]. It follows that if N is not contained in L y , then L ′ is contained in L y , a contradiction. Hence, L y = L , which implies that ∣ y G y ∣ = ∣ y L y ∣ = ∣ y L ∣ = ∣ N ∣ . By the hypotheses and foregoing arguments, N is of order p , a final contradiction. Thus, the proof is complete.□

Proof of Theorem 1.5

We first prove that G is solvable. Since any group of odd order is solvable, we may assume that G is of even order and thus p = 2 . By Lemmas 2.3 and 2.8, we assume that G is not a non-abelian simple group. Let N be a minimal normal subgroup of G . Then, it is easy to see that G / N satisfies our assumption on G by Lemmas 2.1(3) and 2.2(1) provided that 2 ∈ π ( G / N ) and so is solvable by induction. Note that G / N is clearly solvable if 2 ∉ π ( G / N ) . If N is a non-abelian, then Lemma 2.2(3) implies that either N ⩽ A or N ⩽ B and so N is solvable by induction. It follows that G is solvable.

Now we prove that G is p -nilpotent. Suppose that this is false and let G be a counterexample of minimal order. Let N be any minimal normal subgroup of G . Then, by the choice of G , G / N is p -nilpotent since G / N satisfies the conditions. Moreover, we may assume that N is the unique minimal normal subgroup of G as in the proof of Theorem 1.4. If Φ ( G ) is nontrivial, then N is contained in Φ ( G ) and so G / Φ ( G ) is p -nilpotent, which implies that G is also p -nilpotent by [13, Ch. A, Lemma 13.2]. Therefore, Φ ( G ) = 1 and N = C G ( N ) . Discussing as in the last paragraph of the proof of Theorem 1.4, we obtain that N is cyclic of order p , which implies that G / N = G / C G ( N ) is isomorphic to a subgroup of Aut ( N ) ≃ C p − 1 , where C p − 1 is a cyclic group of order p − 1 . This contradicts ( p − 1 , ∣ G ∣ ) = 1 , which finishes the proof.□

Acknowledgments

The authors are very grateful to the referees’ helpful comments and suggestions.

  1. Funding information: This project was supported by NSFC (11671063), the Natural Science Foundation of CSTC (cstc2018jcyjAX0060), the Science and Technology Research Program of Chongqing Municipal Education Commission (KJZD-K202001303) and a grant from the Simons Foundation (No 499532, YY).

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

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Received: 2020-10-08
Revised: 2021-02-13
Accepted: 2021-02-15
Published Online: 2021-05-03

© 2021 Jinbao Li and Yong Yang, published by De Gruyter

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

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  62. Construction of a family of non-stationary combined ternary subdivision schemes reproducing exponential polynomials
  63. On the evolutionary bifurcation curves for the one-dimensional prescribed mean curvature equation with logistic type
  64. On intersections of two non-incident subgroups of finite p-groups
  65. Global existence and boundedness in a two-species chemotaxis system with nonlinear diffusion
  66. Finite groups with 4p2q elements of maximal order
  67. Positive solutions of a discrete nonlinear third-order three-point eigenvalue problem with sign-changing Green's function
  68. Power moments of automorphic L-functions related to Maass forms for SL3(ℤ)
  69. Entire solutions for several general quadratic trinomial differential difference equations
  70. Strong consistency of regression function estimator with martingale difference errors
  71. Fractional Hermite-Hadamard-type inequalities for interval-valued co-ordinated convex functions
  72. Montgomery identity and Ostrowski-type inequalities via quantum calculus
  73. Universal inequalities of the poly-drifting Laplacian on smooth metric measure spaces
  74. On reducible non-Weierstrass semigroups
  75. so-metrizable spaces and images of metric spaces
  76. Some new parameterized inequalities for co-ordinated convex functions involving generalized fractional integrals
  77. The concept of cone b-Banach space and fixed point theorems
  78. Complete consistency for the estimator of nonparametric regression model based on m-END errors
  79. A posteriori error estimates based on superconvergence of FEM for fractional evolution equations
  80. Solution of integral equations via coupled fixed point theorems in 𝔉-complete metric spaces
  81. Symmetric pairs and pseudosymmetry of Θ-Yetter-Drinfeld categories for Hom-Hopf algebras
  82. A new characterization of the automorphism groups of Mathieu groups
  83. The role of w-tilting modules in relative Gorenstein (co)homology
  84. Primitive and decomposable elements in homology of ΩΣℂP∞
  85. The G-sequence shadowing property and G-equicontinuity of the inverse limit spaces under group action
  86. Classification of f-biharmonic submanifolds in Lorentz space forms
  87. Some new results on the weaving of K-g-frames in Hilbert spaces
  88. Matrix representation of a cross product and related curl-based differential operators in all space dimensions
  89. Global optimization and applications to a variational inequality problem
  90. Functional equations related to higher derivations in semiprime rings
  91. A partial order on transformation semigroups with restricted range that preserve double direction equivalence
  92. On multi-step methods for singular fractional q-integro-differential equations
  93. Compact perturbations of operators with property (t)
  94. Entire solutions for several complex partial differential-difference equations of Fermat type in ℂ2
  95. Random attractors for stochastic plate equations with memory in unbounded domains
  96. On the convergence of two-step modulus-based matrix splitting iteration method
  97. On the separation method in stochastic reconstruction problem
  98. Robust estimation for partial functional linear regression models based on FPCA and weighted composite quantile regression
  99. Structure of coincidence isometry groups
  100. Sharp function estimates and boundedness for Toeplitz-type operators associated with general fractional integral operators
  101. Oscillatory hyper-Hilbert transform on Wiener amalgam spaces
  102. Euler-type sums involving multiple harmonic sums and binomial coefficients
  103. Poly-falling factorial sequences and poly-rising factorial sequences
  104. Geometric approximations to transition densities of Jump-type Markov processes
  105. Multiple solutions for a quasilinear Choquard equation with critical nonlinearity
  106. Bifurcations and exact traveling wave solutions for the regularized Schamel equation
  107. Almost factorizable weakly type B semigroups
  108. The finite spectrum of Sturm-Liouville problems with n transmission conditions and quadratic eigenparameter-dependent boundary conditions
  109. Ground state sign-changing solutions for a class of quasilinear Schrödinger equations
  110. Epi-quasi normality
  111. Derivative and higher-order Cauchy integral formula of matrix functions
  112. Commutators of multilinear strongly singular integrals on nonhomogeneous metric measure spaces
  113. Solutions to a multi-phase model of sea ice growth
  114. Existence and simulation of positive solutions for m-point fractional differential equations with derivative terms
  115. Bernstein-Walsh type inequalities for derivatives of algebraic polynomials in quasidisks
  116. Review Article
  117. Semiprimeness of semigroup algebras
  118. Special Issue on Problems, Methods and Applications of Nonlinear Analysis (Part II)
  119. Third-order differential equations with three-point boundary conditions
  120. Fractional calculus, zeta functions and Shannon entropy
  121. Uniqueness of positive solutions for boundary value problems associated with indefinite ϕ-Laplacian-type equations
  122. Synchronization of Caputo fractional neural networks with bounded time variable delays
  123. On quasilinear elliptic problems with finite or infinite potential wells
  124. Deterministic and random approximation by the combination of algebraic polynomials and trigonometric polynomials
  125. On a fractional Schrödinger-Poisson system with strong singularity
  126. Parabolic inequalities in Orlicz spaces with data in L1
  127. Special Issue on Evolution Equations, Theory and Applications (Part II)
  128. Impulsive Caputo-Fabrizio fractional differential equations in b-metric spaces
  129. Existence of a solution of Hilfer fractional hybrid problems via new Krasnoselskii-type fixed point theorems
  130. On a nonlinear system of Riemann-Liouville fractional differential equations with semi-coupled integro-multipoint boundary conditions
  131. Blow-up results of the positive solution for a class of degenerate parabolic equations
  132. Long time decay for 3D Navier-Stokes equations in Fourier-Lei-Lin spaces
  133. On the extinction problem for a p-Laplacian equation with a nonlinear gradient source
  134. General decay rate for a viscoelastic wave equation with distributed delay and Balakrishnan-Taylor damping
  135. On hyponormality on a weighted annulus
  136. Exponential stability of Timoshenko system in thermoelasticity of second sound with a memory and distributed delay term
  137. Convergence results on Picard-Krasnoselskii hybrid iterative process in CAT(0) spaces
  138. Special Issue on Boundary Value Problems and their Applications on Biosciences and Engineering (Part I)
  139. Marangoni convection in layers of water-based nanofluids under the effect of rotation
  140. A transient analysis to the M(τ)/M(τ)/k queue with time-dependent parameters
  141. Existence of random attractors and the upper semicontinuity for small random perturbations of 2D Navier-Stokes equations with linear damping
  142. Degenerate binomial and Poisson random variables associated with degenerate Lah-Bell polynomials
  143. Special Issue on Fractional Problems with Variable-Order or Variable Exponents (Part I)
  144. On the mixed fractional quantum and Hadamard derivatives for impulsive boundary value problems
  145. The Lp dual Minkowski problem about 0 < p < 1 and q > 0
https://doi.org/10.1515/math-2021-0030
Keywords for this article
finite groups; sub-class sizes; mutually permutable products
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  69. Entire solutions for several general quadratic trinomial differential difference equations
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  72. Montgomery identity and Ostrowski-type inequalities via quantum calculus
  73. Universal inequalities of the poly-drifting Laplacian on smooth metric measure spaces
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  75. so-metrizable spaces and images of metric spaces
  76. Some new parameterized inequalities for co-ordinated convex functions involving generalized fractional integrals
  77. The concept of cone b-Banach space and fixed point theorems
  78. Complete consistency for the estimator of nonparametric regression model based on m-END errors
  79. A posteriori error estimates based on superconvergence of FEM for fractional evolution equations
  80. Solution of integral equations via coupled fixed point theorems in 𝔉-complete metric spaces
  81. Symmetric pairs and pseudosymmetry of Θ-Yetter-Drinfeld categories for Hom-Hopf algebras
  82. A new characterization of the automorphism groups of Mathieu groups
  83. The role of w-tilting modules in relative Gorenstein (co)homology
  84. Primitive and decomposable elements in homology of ΩΣℂP∞
  85. The G-sequence shadowing property and G-equicontinuity of the inverse limit spaces under group action
  86. Classification of f-biharmonic submanifolds in Lorentz space forms
  87. Some new results on the weaving of K-g-frames in Hilbert spaces
  88. Matrix representation of a cross product and related curl-based differential operators in all space dimensions
  89. Global optimization and applications to a variational inequality problem
  90. Functional equations related to higher derivations in semiprime rings
  91. A partial order on transformation semigroups with restricted range that preserve double direction equivalence
  92. On multi-step methods for singular fractional q-integro-differential equations
  93. Compact perturbations of operators with property (t)
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  95. Random attractors for stochastic plate equations with memory in unbounded domains
  96. On the convergence of two-step modulus-based matrix splitting iteration method
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  99. Structure of coincidence isometry groups
  100. Sharp function estimates and boundedness for Toeplitz-type operators associated with general fractional integral operators
  101. Oscillatory hyper-Hilbert transform on Wiener amalgam spaces
  102. Euler-type sums involving multiple harmonic sums and binomial coefficients
  103. Poly-falling factorial sequences and poly-rising factorial sequences
  104. Geometric approximations to transition densities of Jump-type Markov processes
  105. Multiple solutions for a quasilinear Choquard equation with critical nonlinearity
  106. Bifurcations and exact traveling wave solutions for the regularized Schamel equation
  107. Almost factorizable weakly type B semigroups
  108. The finite spectrum of Sturm-Liouville problems with n transmission conditions and quadratic eigenparameter-dependent boundary conditions
  109. Ground state sign-changing solutions for a class of quasilinear Schrödinger equations
  110. Epi-quasi normality
  111. Derivative and higher-order Cauchy integral formula of matrix functions
  112. Commutators of multilinear strongly singular integrals on nonhomogeneous metric measure spaces
  113. Solutions to a multi-phase model of sea ice growth
  114. Existence and simulation of positive solutions for m-point fractional differential equations with derivative terms
  115. Bernstein-Walsh type inequalities for derivatives of algebraic polynomials in quasidisks
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  117. Semiprimeness of semigroup algebras
  118. Special Issue on Problems, Methods and Applications of Nonlinear Analysis (Part II)
  119. Third-order differential equations with three-point boundary conditions
  120. Fractional calculus, zeta functions and Shannon entropy
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  122. Synchronization of Caputo fractional neural networks with bounded time variable delays
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  126. Parabolic inequalities in Orlicz spaces with data in L1
  127. Special Issue on Evolution Equations, Theory and Applications (Part II)
  128. Impulsive Caputo-Fabrizio fractional differential equations in b-metric spaces
  129. Existence of a solution of Hilfer fractional hybrid problems via new Krasnoselskii-type fixed point theorems
  130. On a nonlinear system of Riemann-Liouville fractional differential equations with semi-coupled integro-multipoint boundary conditions
  131. Blow-up results of the positive solution for a class of degenerate parabolic equations
  132. Long time decay for 3D Navier-Stokes equations in Fourier-Lei-Lin spaces
  133. On the extinction problem for a p-Laplacian equation with a nonlinear gradient source
  134. General decay rate for a viscoelastic wave equation with distributed delay and Balakrishnan-Taylor damping
  135. On hyponormality on a weighted annulus
  136. Exponential stability of Timoshenko system in thermoelasticity of second sound with a memory and distributed delay term
  137. Convergence results on Picard-Krasnoselskii hybrid iterative process in CAT(0) spaces
  138. Special Issue on Boundary Value Problems and their Applications on Biosciences and Engineering (Part I)
  139. Marangoni convection in layers of water-based nanofluids under the effect of rotation
  140. A transient analysis to the M(τ)/M(τ)/k queue with time-dependent parameters
  141. Existence of random attractors and the upper semicontinuity for small random perturbations of 2D Navier-Stokes equations with linear damping
  142. Degenerate binomial and Poisson random variables associated with degenerate Lah-Bell polynomials
  143. Special Issue on Fractional Problems with Variable-Order or Variable Exponents (Part I)
  144. On the mixed fractional quantum and Hadamard derivatives for impulsive boundary value problems
  145. The Lp dual Minkowski problem about 0 < p < 1 and q > 0
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