Home Finite groups whose all second maximal subgroups are cyclic
Article Open Access

Finite groups whose all second maximal subgroups are cyclic

  • Li Ma , Wei Meng EMAIL logo and Wanqing Ma
Published/Copyright: May 20, 2017
Download Article (PDF)
Open Mathematics
From the journal Open Mathematics Volume 15 Issue 1

Abstract

In this paper, we give a complete classification of the finite groups G whose second maximal subgroups are cyclic

Keywords: Cyclic subgroups; 2-maximal subgroups; Solvable groups
MSC 2010: 20D10; 20D20

1 Introduction

All groups considered in this paper are finite. Throughout the following, G always denotes a finite group. The symbol ￗ(G) denotes the set of the prime divisors of |G|. In 1903, Miller and Moreno [1] gave a complete classification of finite groups in which all maximal subgroups are abelian. In 1924, Shmidt [2] described finite groups whose maximal subgroups are all nilpotent. Suzuki [3] and Janko [4] have described finite unsolvable groups whose 2- maximal subgroups are nilpotent. There are only two such groups: A5 and the special linear group SL.(2,5). In1968, V. A. Belonogov [5] described finite solvable groups whose 2-maximal subgroups are all nilpotent. In 1979, De Vivo[6] investigated finite groups whose 2-maximal subgroups are all Sylow tower groups. In 1988, S.R. Li [7] investigatedfinite unsolvable groups whose all 2-maximal 3d-subgroups are super solvable.

The aim of this paper is to describe finite groups whose second maximal subgroups are all cyclic. For convenience, weintroduce the definition as follows:

Definition 1.1

A group G is called an SMC-group if every second maximal subgroup of G is cyclic.

All unexplained notations and terminologies are standard and can be found in [8-10]

2 Main results

For the proof of the Main Theorem, we need some known results. Below we give the result of Janko, Miller and Moreno

Lemma 2.1 ([4])

Let G be an unsolvable group. If every second maximal subgroup of G is nilpotent, then G is isomorphic to A5 or SL.2,5)

Lemma 2.2 ([1])

Let G be a non-cyclic group all of whose proper subgroups are cyclic. Then one of thefollowing holds:

  1. G ≅ Zp × Zp, p is a prime.

  2. G ≅ Q8.

  3. G≅〈a,b:ap=bqm=1,b−1ab=as〉, where s ≢ 1(mod p), sq ≡ 1(mod p), p and q are distinct primes.

The following Theorem shows that SMC-groups are solvable.

Theorem 2.3

Let G be a non-cyclic SMC-group. Then G is solvable and |ￗ(G)| ≤ 3.

Proof

Suppose G is an unsolvable S M C-group, then all second maximal subgroups of G are cyclic and hence they are nilpotent. By Lemma 2.1, we know G ≅ A5 or SL(2,5). Since each of the groups A5 and SL(2,5) possesses one non-cyclic second maximal subgroup, we have that G is solvable

Let G be a solvable non-cyclic group having cyclic 2-maximal subgroups. Then every proper subgroup of G is a cyclic group or a minimal non-cyclic group, and each minimal non-cyclic group is maximal in G. Because minimal non-cyclic groups satisfy the thesis of Lemma 2.2, we can assume that one maximal subgroup M of G is a minimal non-cyclic group, and since the index of evert maximal subgroup in a solvable group is a power of a prime, we have that |π(G)| ≤3.   □

Corollary 2.4

Let G be an SMC-group. If |π(G)| ≥4, then G is cyclic.

By Corollary 2.4, we only determine the structure of SMC-groups G with |π (G)| ≤3. Firstly, we show the structure of SMC-groups with |π(G)| = 3.

Lemma 2.5

Let G be a non-cyclic SMC-group with |π(G)| = 3. Then all Sylow subgroups of G are cyclic.

Proof

Let G be a non-cyclic SMC-group and π (G) = {p1, p2, p3}, where p, q and r are distinct primes. By Theorem 2.3, we know that G is solvable and hence G possesses a Sylow system {P1, P2, P3}, where Pi ∈ SylPi (G). Thus, Pi < Pi Pj < G for all i ≠ j. Since every 2-maximal subgroup of G is cyclic, we get each Pi is cyclic for i = 1,2,3.   □

A famous result of Burnside, Hölder and Zassenhaus is recalled below.

Lemma 2.6

For an odd m ≥ 1 and an arbitrary n ≥ 1 such that rn ≡1 mod m, 1 ≤ r ≤ m and gcd(n(r−1), m) = 1 , the group

M(m.n)=a,b|am=bn=1,b−1ab=ar

is meta-cyclic and all its Sylow subgroups are cyclic. Conversely, each group with such a property has a presentation of the form of M(m.n).

Suppose that G is a non-cyclic SMC-group with |π (G)| = 3, then G is a meta-cyclic group by Lemma 2.6. Furthermore, the following results hold

Theorem 2.7

Let G be a non-cyclic group with |π(G)| = 3. If G is an SMC-group, then one of the following statements holds.

  1. G = 〈a,b,c 〉, where apm=bq=cr=[b,c]=1, ab = as, ac = at, s ≢ 1 (modq), sp ≡ 1 (modq), t ≢ 1(mod r), tp ≡ 1 (mod r), p, q and r are distinct primes.

  2. G = H × Zr, where H≅〈a,b:apm=bq=1, a-1ba = bs, s ≢ 1(mod q), sp ≡ 1 (mod q) 〉 , p, q and r are distinct primes.

  3. G = 〈 a,b,c 〉 , where ap=bqm=cr=[a,b]=[a,c]=1, ac = as, s ≢ 1(mod r), sq ≡ 1(mod r), p, q and r are distinct primes.

  4. G = 〈 a,b,c 〉, where apm=bq=cr=[a,b]=[b,c]=1, ac = as, s ≡l(mod q),sp ≡ 1 (mod q), p, q and r are distinct primes.

  5. G = 〈 a,b,c 〉, where apm=bq=cr=[a,b]=1, ac = as, bc = bt, s ≢l.mod q), sp = 1 (mod q), t≢ l. mod r), tq ≡ 1 (mod r) p, q and r are distinct primes.

Proof

Suppose that G is a non-cyclic SMC-group and π (G) = {p,q, r}(p < q < r).As G is solvable, we know thatG possesses a Sylow system {P, Q, R}. Assume that P = 〈a〉, Q = 〈b〉 and R = 〈c〉.

Firstly, suppose [P,Q] ≠ l,then PQ is non-cyclic and hence PQ is a maximal subgroup of G and δ(PQ) = 1. By Lemma 2.2, we get that PQ=〈a,b:apm=bq=1,ba=bs〉.. Suppose that [P, R] ≠ l. As in the above argument, then PR=〈a,c:apm=cr=1,ca=ct〉. By Lemma 2.6, we get [Q,R] = 1 and hence the conclusion (1) holds.

Assume [P, R] = 1. Then PR is a cyclic group. We claim that [Q, R] = l. Suppose that [Q, R] ≠l, we get one of Q and PR is normal in G by Lemma 2.6. If PR is normal in G, then P isnormal in G, which is contrary to [P, Q] ≠ 1. Thus, Q is normal in G. In another, R is normal in G. Hence we have QR = Q × R, that is, [Q,R] = l. The conclusion (2) holds.

Secondly, suppose [P, Q] = 1. If [P,R] = 1, then [Q, R] ≠ 1 and QR is a minimal non-cyclic group. Thus, we get the conclusion (3). In the following, suppose [P, R] ≠ 1 Similarly to the above argument, we get the conclusion(4)and(5)

The following Theorem shows the structure of SMC-groups G with |π(G)| = 2

Theorem 2.8

Let G be a non-cyclic group with |π(G)| = 2. If G is an SMC-group, then one of the following statements hold.

  1. G is a minimal non-cyclic group.

  2. G (Zp × Zp)Zq, where Zp × Zp ⊴ G, p and q are distinct primes.

  3. G = Q8 × Zp, p is an odd prime.

  4. G = Q8 × Z3.

  5. G = 〈a,b〉, where ap=bqm=1, b-1 ab = as, sq ≢ l(mod p), sq2 ≡ (mod p), m ≥ 2, p and q are distinct primes.

  6. G = 〈a,b,c〉, where ap = b2 = [a,b] = 1, b2 = c2, b-1cb = c-1, c-1ac = cat, t ≢ l(mod p), t2 ≡ 1(mod p).

  7. G ≅ 〈a, b〉, where ap2=bqm=1, b-1ab = at ,tq ≢ l(mod p, t q 2 ≡ 1 ( mod p ) .

Proof

Let G be an SMC-group with π (G) = {p,q}. Since G is solvable, there exists a maximal subgroup M of G such that M is normal in G. Hence |G : M| is a prime. Suppose that |G : M| = q, then there exists a q-element c such that G = M〈c〉 with cq ∈ M. Suppose G is not a minimal non-cyclic group, then every maximal subgroup of G is either a cyclic group or a minimal non-cyclic group. Thus, we need to treat the following two cases for M.

Case I: M is a minimal non-cyclic group.

By Lemma 2.2, we need to treat the following three cases for M

  1. M ≅ Zp × Zp.

    Since G is not a p-group, we have G = MZq for some prime q(≠ p). Thus, G proves to be a group of type (2).

  2. M ≅ Q8.

    In this case, G = Q8 Zq, where Q8 ⊴ G and q is an odd prime. If G = Q8 × Zq, then we get conclusion (3). Suppose Zq is not normal in G, then Zq induces an automorphism of Q8 of order q. We know that Aut(Q8) ≅ S4. Hence q = 3,which gives conclusion (4): G = Q8 × Z3.

  3. M = 〈a,b〉 ap=bqm=1, b-1ab = as, s ≢ 1(modp), sq ≡ 1 (mod p)

    In this case, let H = 〈a〉 be normal of order p in G and a ∉ Z(G). Thus CG(H) < G. Moreover, G/CG(H) is cyclic of order dividing p − 1. Hence G/CG(H) is a cyclic q-group. Since CG(H) ≠ M, we see that CG(H) is cyclic. Also, as CG(H) ⊴ G, we can assume that CG(H) is not maximal in G. Consequently |G/CG(H)| = qt, t ≥ 2. Moreover, by the definition of M, we have bq ∈ CG(H) but b ∉ CG(H). This shows that 〈b〉CG(H) is non-abelian of index qt-1. As M = 〈a, b〉 ⊆ 〈 b〉 CG(H), we get that 〈b〉 CG(H) = M and hence t = 2. Now, G/CG(H) is cyclic of order q2 and the Sylow q-subgroup Q = 〈x〉 of G is cyclic of order qm+1, xq2 ∈ CG(H) but xq ∉ CG(H). The above argument implies conclusion(5).

    G = 〈 a , x : a p = x q m = 1 , x − 1 a x = a s , s q ≢ 1 ( mod p ) , s q 2 ≡ ( mod p ) 〉 , m ≥ 2.

Case II: M is a cyclic group.

  1. π(M) = π(G).

    Write |M| = pm1 qm2. We consider the Sylow decomposition of M as

    M = a × b , where a p m 1 = b q m 2 = 1.

    Firstly, we suppose that [a, c] = [b, c] = 1. Then G is abelian. Hence H = 〈b, c〉 is a non-cyclic proper subgroup of order a power of q. So H is a maximal subgroup of G and is a minimal non-cyclic subgroup. By Lemma 2.2, H≅ Zq × Zq and hence G ≅ Zq × Zq × Zp. Thus, G proves to be a group of type (2).

    Secondly, suppose that G is non-abelian.

    Assume that [b, c] ≠ 1. Consider the non-abelian subgroup H = 〈b, c〉, then H is a maximal subgroup of G of order a power of q. By Lemma 2.2, we see that H ≅ Q8 and q = 2. If [a, c] = 1, then G ≅ Q8 × Zp, which yields conclusion (3).

    Assume that [a, c] ≠ 1. Set K = 〈a, c〉 ,then K is a non-cyclic subgroup of G and hence H is a minimal noncyclic group. By Lemma 2.2, we get K = a , c : c 4 = a p = 1 , c − 1 a c = a t , t ≢ 1 ( mod p ) , s 2 ≡ 1 ( mod q ) .

    G = a , b , c : a p = b 2 = [ a , b ] = 1 , b 2 = c 2 , b − 1 c b = c − 1 , c − 1 a c = c a t , t ≢ 1 ( mod p ) , t 2 ≡ 1 ( mod p ) .

    Therefore G is of type (6).

  2. |M| = pn.

    Let M = 〈a〉 ,where o(a) = pn. Then the G = 〈a, b〉 is a non-abelian group, where 〈a〉 is normal in G, 〈b〉 is non-normal with order qm, n ≥ 1, m ≥ 1.By a simple theorem, we know that q < p and so p is an odd prime. nn-1 Thus b as an automorphism of 〈 a〉 is fixed point free. If n ≥ 2, then the subgroup B generalized by apn-1and b is non-abelian of order pqm. Hence B is a minimal non-cyclic group. By Lemma 2.2, we know that n = 2. and we obtain

    G ≅ 〈 a , b : a p 2 = b q m = 1 , , b − 1 a b = a t , t q ≢ 1 ( mod p ) , t q 2 ≡ ( mod p ) 〉 .

    Therefore G is of type (7).

The proof is now complete.

To determine the structure of SMC-p-groups, we need the following known results.

Lemma 2.9 ([11])

Let G be a group of order 24 and M ≅ Q8 be a maximal subgroup of G. Then one of the following statements is true:

  1. G ≅ Q16, a generalized quaternion 2-group of order 24.

  2. G ≅ Q8 × Z2.

  3. G ≅ Q8 * Z4, where * denotes a central product.

Theorem 2.10

Let G be a p-group. Then G is a non-cyclic SMC-group if and only if either |G| ≤ p or G ≅ Q16.

Proof

If all maximal subgroups of G are cyclic, then G is a minimal non-cyclic p-group. By Lemma 2.2, we knowthat G is isomorphic to Q8 or Zp × p, hence |G| ≤ p3. Let M be a non-cyclic maximal subgroup of G, then M is a minimal non-cyclic group and hence |M| ≤ p3. So we have |G| ≤ p4. If |G| = p4, then |M| = p3. By Lemma 2.2, we know M ≅ Q8. Hence G is a group of order 24 and M ≅Q8 is a maximal subgroup of G. By Lemma 2.9, we know that G is one of the groups Q16, Q8 × Z2 or Q8 * Z4. It can be easily shown that both Q8 × Z2 and Q8 * Z4> have a non-cyclic 2-maximal subgroups. Thus, we have G ≅ Q16. The proof is now complete.   □

Theorem

Let G be a non-cyclic SMC-group. Then G is solvable and |π(G)| ≤3. Furthermore, one of the following statements is true:

  1. G is a minimal non-cyclic group.

  2. G is a p-group of order p3.

  3. G is a generalized quaternion 2-group of order 24.

  4. G = (Zp × Zp) Zq, where Zp × Zp ⊴ G, p and q are distinct primes.

  5. G = Q8 × Zp, p is an odd prime.

  6. a , b , w h e r e a p 2 = b q m = 1 , b − 1 a b = a s , s ≢ 1 ( mod p 2 ) , s q ≡ 1 ( mod p 2 ) , p and q are distinct primes.

  7. G = Q8 × Z3.

  8. G = 〈a, b〉, where a p = b q m = 1 , b − 1 a b = a s , s q ≢ 1 ( mod p ) , s q 2 ≡ 1 ( mod p ) , m ≥ 2, p and q are distinct primes.

  9. G = a , b , c w h e r e a p = b 2 = [ a , b ] = 1 , b 2 = c 2 , b − 1 c b = c − 1 a c = c a t , t ≢ 1 ( mod p ) , t 2 ≡ 1 ( mod p ) .

  10. G = 〈a,b,c〉, where apm=bq=cr=[b,c]=1, ab = as, ac = at, s ≢ 1(mod q), sp ≡ 1 (mod q), t ≢ l(mod r), tp ≡ 1 (mod r ), p, q and r are distinct primes.

  11. G = H × Zr, where H ≅ 〈 a , b : a p m = b q = 1 , a − 1 b a = b s , s ≢ 1 ( mod q ) , s p ≡ 1 ( mod q ) 〉 , p, q and r are distinct primes.

  12. G = 〈a,b,c〉, where ap=bqm=cr=[a,b]=[a,c]=1, ac = as, s≢ l(mod r), sq ≡ 1 (mod r), p, q and r are distinct primes.

  13. G = 〈a,b,c〉, where apm=bq=cr=[a,b]=[b,c]=1, ac = as, s≢ l(mod q), sp ≡ 1 (mod q), p, q and r are distinct primes.

  14. G = 〈a,b,c〉, where apm=bq=cr=[a,b]=1, ac = as, bc = bt, s ≢ l(mod q), sp ≡ 1 (mod q), t ≢ l(modr), tq ≡ 1 (mod r) p, q and r are distinct primes.

Proof

The proof of Main Theorem comes from the Theorem 2.7, 2.8 and 2.10.   □

Acknowledgement

W. Meng is supported by National Natural Science Foundation of China (11361075)and the Project of Guangxi Colleges and Universities Key Laboratory of Mathematical and Statistical Model(2016GXKLMS002). L. Ma is supported by National Natural Science Foundation of China (11601263).

References

1 Miller G.A. and Moreno H.C., Non-abelian groups in which every subgroup is abelian. Trans. Amer. Math. Soc. 1903,4, 398-404.10.1090/S0002-9947-1903-1500650-9Search in Google Scholar

2 Schmidt O. Yu., Groups whose subgroups are all special, Matem. sb. 1924, 31,366-372.Search in Google Scholar

3 Suzuki M., The nonexistence of a certain type of simple groups of odd order, Proc. Am. Math. Soc. 1957,8, 686-695.10.1090/S0002-9939-1957-0086818-0Search in Google Scholar

4 Janko Z., Endliche Gruppen mit lauter nilpotenten zweitmaximalen Untergruppen, Math. Z. 1962,79,422-424.10.1007/BF01193133Search in Google Scholar

5 Belonogov V. A., Finite solvable groups with nilpotent 2-maximal subgroups, Math. Z. 1968, 3, 21-32.10.1007/BF01386958Search in Google Scholar

6 Devivo C., A minimality problem concerning Sylow towers in finite groups, Rend. Accad. Sci. Fis. Mat. Napoli, 1979,46, 43-61.Search in Google Scholar

7 Li S.R., Finite unsolvable groups whose every 2-maximal 3d-subgroup is super solvable, Chinese Annals of Mathematics(InChinese), 1988, 9A,32-37.Search in Google Scholar

8 Gorenstein D., Finite groups, Harper & Row Pubishers, New York, 1980.Search in Google Scholar

9 Huppert B., Endliche Gruppen I, Springer-Verlag, Berlin, Heidelberg, New York, 1967.10.1007/978-3-642-64981-3Search in Google Scholar

10 D. J. S. Robinson, A Course in the Theory of Groups. 2nd ed. New York: Springer-Verlag, 1996.Search in Google Scholar

11 M. Y. Xu and H. P Qu, Finite p-groups, Peking University Press(in Chinese), 2010.Search in Google Scholar
Received: 2016-9-20
Accepted: 2017-3-21
Published Online: 2017-5-20

© 2017 Ma et al.

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License.

Articles in the same Issue

  1. Regular Articles
  2. Integrals of Frullani type and the method of brackets
  3. Regular Articles
  4. Edge of chaos in reaction diffusion CNN model
  5. Regular Articles
  6. Calculus using proximities: a mathematical approach in which students can actually prove theorems
  7. Regular Articles
  8. An investigation on hyper S-posets over ordered semihypergroups
  9. Regular Articles
  10. The Leibniz algebras whose subalgebras are ideals
  11. Regular Articles
  12. Fixed point and multidimensional fixed point theorems with applications to nonlinear matrix equations in terms of weak altering distance functions
  13. Regular Articles
  14. Matrix rank and inertia formulas in the analysis of general linear models
  15. Regular Articles
  16. The hybrid power mean of quartic Gauss sums and Kloosterman sums
  17. Regular Articles
  18. Tauberian theorems for statistically (C,1,1) summable double sequences of fuzzy numbers
  19. Regular Articles
  20. Some properties of graded comultiplication modules
  21. Regular Articles
  22. The characterizations of upper approximation operators based on special coverings
  23. Regular Articles
  24. Bi-integrable and tri-integrable couplings of a soliton hierarchy associated with SO(4)
  25. Regular Articles
  26. Dynamics for a discrete competition and cooperation model of two enterprises with multiple delays and feedback controls
  27. Regular Articles
  28. A new view of relationship between atomic posets and complete (algebraic) lattices
  29. Regular Articles
  30. A class of extensions of Restricted (s, t)-Wythoff’s game
  31. Regular Articles
  32. New bounds for the minimum eigenvalue of 𝓜-tensors
  33. Regular Articles
  34. Shintani and Shimura lifts of cusp forms on certain arithmetic groups and their applications
  35. Regular Articles
  36. Empirical likelihood for quantile regression models with response data missing at random
  37. Regular Articles
  38. Convex combination of analytic functions
  39. Regular Articles
  40. On the Yang-Baxter-like matrix equation for rank-two matrices
  41. Regular Articles
  42. Uniform topology on EQ-algebras
  43. Regular Articles
  44. Integrations on rings
  45. Regular Articles
  46. The quasilinear parabolic kirchhoff equation
  47. Regular Articles
  48. Avoiding rainbow 2-connected subgraphs
  49. Regular Articles
  50. On non-Hopfian groups of fractions
  51. Regular Articles
  52. Singularly perturbed hyperbolic problems on metric graphs: asymptotics of solutions
  53. Regular Articles
  54. Rings in which elements are the sum of a nilpotent and a root of a fixed polynomial that commute
  55. Regular Articles
  56. Superstability of functional equations related to spherical functions
  57. Regular Articles
  58. Evaluation of the convolution sum involving the sum of divisors function for 22, 44 and 52
  59. Regular Articles
  60. Weighted minimal translation surfaces in the Galilean space with density
  61. Regular Articles
  62. Complete convergence for weighted sums of pairwise independent random variables
  63. Regular Articles
  64. Binomials transformation formulae for scaled Fibonacci numbers
  65. Regular Articles
  66. Growth functions for some uniformly amenable groups
  67. Regular Articles
  68. Hopf bifurcations in a three-species food chain system with multiple delays
  69. Regular Articles
  70. Oscillation and nonoscillation of half-linear Euler type differential equations with different periodic coefficients
  71. Regular Articles
  72. Osculating curves in 4-dimensional semi-Euclidean space with index 2
  73. Regular Articles
  74. Some new facts about group 𝒢 generated by the family of convergent permutations
  75. Regular Articles
  76. lnfinitely many solutions for fractional Schrödinger equations with perturbation via variational methods
  77. Regular Articles
  78. Supersolvable orders and inductively free arrangements
  79. Regular Articles
  80. Asymptotically almost automorphic solutions of differential equations with piecewise constant argument
  81. Regular Articles
  82. Finite groups whose all second maximal subgroups are cyclic
  83. Regular Articles
  84. Semilinear systems with a multi-valued nonlinear term
  85. Regular Articles
  86. Positive solutions for Hadamard differential systems with fractional integral conditions on an unbounded domain
  87. Regular Articles
  88. Calibration and simulation of Heston model
  89. Regular Articles
  90. One kind sixth power mean of the three-term exponential sums
  91. Regular Articles
  92. Cyclic pairs and common best proximity points in uniformly convex Banach spaces
  93. Regular Articles
  94. The uniqueness of meromorphic functions in k-punctured complex plane
  95. Regular Articles
  96. Normalizers of intermediate congruence subgroups of the Hecke subgroups
  97. Regular Articles
  98. The hyperbolicity constant of infinite circulant graphs
  99. Regular Articles
  100. Scott convergence and fuzzy Scott topology on L-posets
  101. Regular Articles
  102. One sided strong laws for random variables with infinite mean
  103. Regular Articles
  104. The join of split graphs whose completely regular endomorphisms form a monoid
  105. Regular Articles
  106. A new branch and bound algorithm for minimax ratios problems
  107. Regular Articles
  108. Upper bound estimate of incomplete Cochrane sum
  109. Regular Articles
  110. Value distributions of solutions to complex linear differential equations in angular domains
  111. Regular Articles
  112. The nonlinear diffusion equation of the ideal barotropic gas through a porous medium
  113. Regular Articles
  114. The Sheffer stroke operation reducts of basic algebras
  115. Regular Articles
  116. Extensions and improvements of Sherman’s and related inequalities for n-convex functions
  117. Regular Articles
  118. Classification lattices are geometric for complete atomistic lattices
  119. Regular Articles
  120. Possible numbers of x’s in an {x, y}-matrix with a given rank
  121. Regular Articles
  122. New error bounds for linear complementarity problems of weakly chained diagonally dominant B-matrices
  123. Regular Articles
  124. Boundedness of vector-valued B-singular integral operators in Lebesgue spaces
  125. Regular Articles
  126. On the Golomb’s conjecture and Lehmer’s numbers
  127. Regular Articles
  128. Some applications of the Archimedean copulas in the proof of the almost sure central limit theorem for ordinary maxima
  129. Regular Articles
  130. Dual-stage adaptive finite-time modified function projective multi-lag combined synchronization for multiple uncertain chaotic systems
  131. Regular Articles
  132. Corrigendum to: Dual-stage adaptive finite-time modified function projective multi-lag combined synchronization for multiple uncertain chaotic systems
  133. Regular Articles
  134. Convergence and stability of generalized φ-weak contraction mapping in CAT(0) spaces
  135. Regular Articles
  136. Triple solutions for a Dirichlet boundary value problem involving a perturbed discrete p(k)-Laplacian operator
  137. Regular Articles
  138. OD-characterization of alternating groups Ap+d
  139. Regular Articles
  140. On Jordan mappings of inverse semirings
  141. Regular Articles
  142. On generalized Ehresmann semigroups
  143. Regular Articles
  144. On topological properties of spaces obtained by the double band matrix
  145. Regular Articles
  146. Representing derivatives of Chebyshev polynomials by Chebyshev polynomials and related questions
  147. Regular Articles
  148. Chain conditions on composite Hurwitz series rings
  149. Regular Articles
  150. Coloring subgraphs with restricted amounts of hues
  151. Regular Articles
  152. An extension of the method of brackets. Part 1
  153. Regular Articles
  154. Branch-delete-bound algorithm for globally solving quadratically constrained quadratic programs
  155. Regular Articles
  156. Strong edge geodetic problem in networks
  157. Regular Articles
  158. Ricci solitons on almost Kenmotsu 3-manifolds
  159. Regular Articles
  160. Uniqueness of meromorphic functions sharing two finite sets
  161. Regular Articles
  162. On the fourth-order linear recurrence formula related to classical Gauss sums
  163. Regular Articles
  164. Dynamical behavior for a stochastic two-species competitive model
  165. Regular Articles
  166. Two new eigenvalue localization sets for tensors and theirs applications
  167. Regular Articles
  168. κ-strong sequences and the existence of generalized independent families
  169. Regular Articles
  170. Commutators of Littlewood-Paley gκ∗ -functions on non-homogeneous metric measure spaces
  171. Regular Articles
  172. On decompositions of estimators under a general linear model with partial parameter restrictions
  173. Regular Articles
  174. Groups and monoids of Pythagorean triples connected to conics
  175. Regular Articles
  176. Hom-Lie superalgebra structures on exceptional simple Lie superalgebras of vector fields
  177. Regular Articles
  178. Numerical methods for the multiplicative partial differential equations
  179. Regular Articles
  180. Solvable Leibniz algebras with NFn⊕ Fm1 nilradical
  181. Regular Articles
  182. Evaluation of the convolution sums ∑al+bm=n lσ(l) σ(m) with ab ≤ 9
  183. Regular Articles
  184. A study on soft rough semigroups and corresponding decision making applications
  185. Regular Articles
  186. Some new inequalities of Hermite-Hadamard type for s-convex functions with applications
  187. Regular Articles
  188. Deficiency of forests
  189. Regular Articles
  190. Perfect codes in power graphs of finite groups
  191. Regular Articles
  192. A new compact finite difference quasilinearization method for nonlinear evolution partial differential equations
  193. Regular Articles
  194. Does any convex quadrilateral have circumscribed ellipses?
  195. Regular Articles
  196. The dynamic of a Lie group endomorphism
  197. Regular Articles
  198. On pairs of equations in unlike powers of primes and powers of 2
  199. Regular Articles
  200. Differential subordination and convexity criteria of integral operators
  201. Regular Articles
  202. Quantitative relations between short intervals and exceptional sets of cubic Waring-Goldbach problem
  203. Regular Articles
  204. On θ-commutators and the corresponding non-commuting graphs
  205. Regular Articles
  206. Quasi-maximum likelihood estimator of Laplace (1, 1) for GARCH models
  207. Regular Articles
  208. Multiple and sign-changing solutions for discrete Robin boundary value problem with parameter dependence
  209. Regular Articles
  210. Fundamental relation on m-idempotent hyperrings
  211. Regular Articles
  212. A novel recursive method to reconstruct multivariate functions on the unit cube
  213. Regular Articles
  214. Nabla inequalities and permanence for a logistic integrodifferential equation on time scales
  215. Regular Articles
  216. Enumeration of spanning trees in the sequence of Dürer graphs
  217. Regular Articles
  218. Quotient of information matrices in comparison of linear experiments for quadratic estimation
  219. Regular Articles
  220. Fourier series of functions involving higher-order ordered Bell polynomials
  221. Regular Articles
  222. Simple modules over Auslander regular rings
  223. Regular Articles
  224. Weighted multilinear p-adic Hardy operators and commutators
  225. Regular Articles
  226. Guaranteed cost finite-time control of positive switched nonlinear systems with D-perturbation
  227. Regular Articles
  228. A modified quasi-boundary value method for an abstract ill-posed biparabolic problem
  229. Regular Articles
  230. Extended Riemann-Liouville type fractional derivative operator with applications
  231. Topical Issue on Topological and Algebraic Genericity in Infinite Dimensional Spaces
  232. The algebraic size of the family of injective operators
  233. Topical Issue on Topological and Algebraic Genericity in Infinite Dimensional Spaces
  234. The history of a general criterium on spaceability
  235. Topical Issue on Topological and Algebraic Genericity in Infinite Dimensional Spaces
  236. On sequences not enjoying Schur’s property
  237. Topical Issue on Topological and Algebraic Genericity in Infinite Dimensional Spaces
  238. A hierarchy in the family of real surjective functions
  239. Topical Issue on Topological and Algebraic Genericity in Infinite Dimensional Spaces
  240. Dynamics of multivalued linear operators
  241. Topical Issue on Topological and Algebraic Genericity in Infinite Dimensional Spaces
  242. Linear dynamics of semigroups generated by differential operators
  243. Special Issue on Recent Developments in Differential Equations
  244. Isomorphism theorems for some parabolic initial-boundary value problems in Hörmander spaces
  245. Special Issue on Recent Developments in Differential Equations
  246. Determination of a diffusion coefficient in a quasilinear parabolic equation
  247. Special Issue on Recent Developments in Differential Equations
  248. Homogeneous two-point problem for PDE of the second order in time variable and infinite order in spatial variables
  249. Special Issue on Recent Developments in Differential Equations
  250. A nonlinear plate control without linearization
  251. Special Issue on Recent Developments in Differential Equations
  252. Reduction of a Schwartz-type boundary value problem for biharmonic monogenic functions to Fredholm integral equations
  253. Special Issue on Recent Developments in Differential Equations
  254. Inverse problem for a physiologically structured population model with variable-effort harvesting
  255. Special Issue on Recent Developments in Differential Equations
  256. Existence of solutions for delay evolution equations with nonlocal conditions
  257. Special Issue on Recent Developments in Differential Equations
  258. Comments on behaviour of solutions of elliptic quasi-linear problems in a neighbourhood of boundary singularities
  259. Special Issue on Recent Developments in Differential Equations
  260. Coupled fixed point theorems in complete metric spaces endowed with a directed graph and application
  261. Special Issue on Recent Developments in Differential Equations
  262. Existence of entropy solutions for nonlinear elliptic degenerate anisotropic equations
  263. Special Issue on Recent Developments in Differential Equations
  264. Integro-differential systems with variable exponents of nonlinearity
  265. Special Issue on Recent Developments in Differential Equations
  266. Elliptic operators on refined Sobolev scales on vector bundles
  267. Special Issue on Recent Developments in Differential Equations
  268. Multiplicity solutions of a class fractional Schrödinger equations
  269. Special Issue on Recent Developments in Differential Equations
  270. Determining of right-hand side of higher order ultraparabolic equation
  271. Special Issue on Recent Developments in Differential Equations
  272. Asymptotic approximation for the solution to a semi-linear elliptic problem in a thin aneurysm-type domain
  273. Topical Issue on Metaheuristics - Methods and Applications
  274. Learnheuristics: hybridizing metaheuristics with machine learning for optimization with dynamic inputs
  275. Topical Issue on Metaheuristics - Methods and Applications
  276. Nature–inspired metaheuristic algorithms to find near–OGR sequences for WDM channel allocation and their performance comparison
  277. Topical Issue on Cyber-security Mathematics
  278. Monomial codes seen as invariant subspaces
  279. Topical Issue on Cyber-security Mathematics
  280. Expert knowledge and data analysis for detecting advanced persistent threats
  281. Topical Issue on Cyber-security Mathematics
  282. Feedback equivalence of convolutional codes over finite rings
https://doi.org/10.1515/math-2017-0054
Keywords for this article
Cyclic subgroups; 2-maximal subgroups; Solvable groups
Creative Commons
BY-NC-ND 3.0

Articles in the same Issue

  1. Regular Articles
  2. Integrals of Frullani type and the method of brackets
  3. Regular Articles
  4. Edge of chaos in reaction diffusion CNN model
  5. Regular Articles
  6. Calculus using proximities: a mathematical approach in which students can actually prove theorems
  7. Regular Articles
  8. An investigation on hyper S-posets over ordered semihypergroups
  9. Regular Articles
  10. The Leibniz algebras whose subalgebras are ideals
  11. Regular Articles
  12. Fixed point and multidimensional fixed point theorems with applications to nonlinear matrix equations in terms of weak altering distance functions
  13. Regular Articles
  14. Matrix rank and inertia formulas in the analysis of general linear models
  15. Regular Articles
  16. The hybrid power mean of quartic Gauss sums and Kloosterman sums
  17. Regular Articles
  18. Tauberian theorems for statistically (C,1,1) summable double sequences of fuzzy numbers
  19. Regular Articles
  20. Some properties of graded comultiplication modules
  21. Regular Articles
  22. The characterizations of upper approximation operators based on special coverings
  23. Regular Articles
  24. Bi-integrable and tri-integrable couplings of a soliton hierarchy associated with SO(4)
  25. Regular Articles
  26. Dynamics for a discrete competition and cooperation model of two enterprises with multiple delays and feedback controls
  27. Regular Articles
  28. A new view of relationship between atomic posets and complete (algebraic) lattices
  29. Regular Articles
  30. A class of extensions of Restricted (s, t)-Wythoff’s game
  31. Regular Articles
  32. New bounds for the minimum eigenvalue of 𝓜-tensors
  33. Regular Articles
  34. Shintani and Shimura lifts of cusp forms on certain arithmetic groups and their applications
  35. Regular Articles
  36. Empirical likelihood for quantile regression models with response data missing at random
  37. Regular Articles
  38. Convex combination of analytic functions
  39. Regular Articles
  40. On the Yang-Baxter-like matrix equation for rank-two matrices
  41. Regular Articles
  42. Uniform topology on EQ-algebras
  43. Regular Articles
  44. Integrations on rings
  45. Regular Articles
  46. The quasilinear parabolic kirchhoff equation
  47. Regular Articles
  48. Avoiding rainbow 2-connected subgraphs
  49. Regular Articles
  50. On non-Hopfian groups of fractions
  51. Regular Articles
  52. Singularly perturbed hyperbolic problems on metric graphs: asymptotics of solutions
  53. Regular Articles
  54. Rings in which elements are the sum of a nilpotent and a root of a fixed polynomial that commute
  55. Regular Articles
  56. Superstability of functional equations related to spherical functions
  57. Regular Articles
  58. Evaluation of the convolution sum involving the sum of divisors function for 22, 44 and 52
  59. Regular Articles
  60. Weighted minimal translation surfaces in the Galilean space with density
  61. Regular Articles
  62. Complete convergence for weighted sums of pairwise independent random variables
  63. Regular Articles
  64. Binomials transformation formulae for scaled Fibonacci numbers
  65. Regular Articles
  66. Growth functions for some uniformly amenable groups
  67. Regular Articles
  68. Hopf bifurcations in a three-species food chain system with multiple delays
  69. Regular Articles
  70. Oscillation and nonoscillation of half-linear Euler type differential equations with different periodic coefficients
  71. Regular Articles
  72. Osculating curves in 4-dimensional semi-Euclidean space with index 2
  73. Regular Articles
  74. Some new facts about group 𝒢 generated by the family of convergent permutations
  75. Regular Articles
  76. lnfinitely many solutions for fractional Schrödinger equations with perturbation via variational methods
  77. Regular Articles
  78. Supersolvable orders and inductively free arrangements
  79. Regular Articles
  80. Asymptotically almost automorphic solutions of differential equations with piecewise constant argument
  81. Regular Articles
  82. Finite groups whose all second maximal subgroups are cyclic
  83. Regular Articles
  84. Semilinear systems with a multi-valued nonlinear term
  85. Regular Articles
  86. Positive solutions for Hadamard differential systems with fractional integral conditions on an unbounded domain
  87. Regular Articles
  88. Calibration and simulation of Heston model
  89. Regular Articles
  90. One kind sixth power mean of the three-term exponential sums
  91. Regular Articles
  92. Cyclic pairs and common best proximity points in uniformly convex Banach spaces
  93. Regular Articles
  94. The uniqueness of meromorphic functions in k-punctured complex plane
  95. Regular Articles
  96. Normalizers of intermediate congruence subgroups of the Hecke subgroups
  97. Regular Articles
  98. The hyperbolicity constant of infinite circulant graphs
  99. Regular Articles
  100. Scott convergence and fuzzy Scott topology on L-posets
  101. Regular Articles
  102. One sided strong laws for random variables with infinite mean
  103. Regular Articles
  104. The join of split graphs whose completely regular endomorphisms form a monoid
  105. Regular Articles
  106. A new branch and bound algorithm for minimax ratios problems
  107. Regular Articles
  108. Upper bound estimate of incomplete Cochrane sum
  109. Regular Articles
  110. Value distributions of solutions to complex linear differential equations in angular domains
  111. Regular Articles
  112. The nonlinear diffusion equation of the ideal barotropic gas through a porous medium
  113. Regular Articles
  114. The Sheffer stroke operation reducts of basic algebras
  115. Regular Articles
  116. Extensions and improvements of Sherman’s and related inequalities for n-convex functions
  117. Regular Articles
  118. Classification lattices are geometric for complete atomistic lattices
  119. Regular Articles
  120. Possible numbers of x’s in an {x, y}-matrix with a given rank
  121. Regular Articles
  122. New error bounds for linear complementarity problems of weakly chained diagonally dominant B-matrices
  123. Regular Articles
  124. Boundedness of vector-valued B-singular integral operators in Lebesgue spaces
  125. Regular Articles
  126. On the Golomb’s conjecture and Lehmer’s numbers
  127. Regular Articles
  128. Some applications of the Archimedean copulas in the proof of the almost sure central limit theorem for ordinary maxima
  129. Regular Articles
  130. Dual-stage adaptive finite-time modified function projective multi-lag combined synchronization for multiple uncertain chaotic systems
  131. Regular Articles
  132. Corrigendum to: Dual-stage adaptive finite-time modified function projective multi-lag combined synchronization for multiple uncertain chaotic systems
  133. Regular Articles
  134. Convergence and stability of generalized φ-weak contraction mapping in CAT(0) spaces
  135. Regular Articles
  136. Triple solutions for a Dirichlet boundary value problem involving a perturbed discrete p(k)-Laplacian operator
  137. Regular Articles
  138. OD-characterization of alternating groups Ap+d
  139. Regular Articles
  140. On Jordan mappings of inverse semirings
  141. Regular Articles
  142. On generalized Ehresmann semigroups
  143. Regular Articles
  144. On topological properties of spaces obtained by the double band matrix
  145. Regular Articles
  146. Representing derivatives of Chebyshev polynomials by Chebyshev polynomials and related questions
  147. Regular Articles
  148. Chain conditions on composite Hurwitz series rings
  149. Regular Articles
  150. Coloring subgraphs with restricted amounts of hues
  151. Regular Articles
  152. An extension of the method of brackets. Part 1
  153. Regular Articles
  154. Branch-delete-bound algorithm for globally solving quadratically constrained quadratic programs
  155. Regular Articles
  156. Strong edge geodetic problem in networks
  157. Regular Articles
  158. Ricci solitons on almost Kenmotsu 3-manifolds
  159. Regular Articles
  160. Uniqueness of meromorphic functions sharing two finite sets
  161. Regular Articles
  162. On the fourth-order linear recurrence formula related to classical Gauss sums
  163. Regular Articles
  164. Dynamical behavior for a stochastic two-species competitive model
  165. Regular Articles
  166. Two new eigenvalue localization sets for tensors and theirs applications
  167. Regular Articles
  168. κ-strong sequences and the existence of generalized independent families
  169. Regular Articles
  170. Commutators of Littlewood-Paley gκ∗ -functions on non-homogeneous metric measure spaces
  171. Regular Articles
  172. On decompositions of estimators under a general linear model with partial parameter restrictions
  173. Regular Articles
  174. Groups and monoids of Pythagorean triples connected to conics
  175. Regular Articles
  176. Hom-Lie superalgebra structures on exceptional simple Lie superalgebras of vector fields
  177. Regular Articles
  178. Numerical methods for the multiplicative partial differential equations
  179. Regular Articles
  180. Solvable Leibniz algebras with NFn⊕ Fm1 nilradical
  181. Regular Articles
  182. Evaluation of the convolution sums ∑al+bm=n lσ(l) σ(m) with ab ≤ 9
  183. Regular Articles
  184. A study on soft rough semigroups and corresponding decision making applications
  185. Regular Articles
  186. Some new inequalities of Hermite-Hadamard type for s-convex functions with applications
  187. Regular Articles
  188. Deficiency of forests
  189. Regular Articles
  190. Perfect codes in power graphs of finite groups
  191. Regular Articles
  192. A new compact finite difference quasilinearization method for nonlinear evolution partial differential equations
  193. Regular Articles
  194. Does any convex quadrilateral have circumscribed ellipses?
  195. Regular Articles
  196. The dynamic of a Lie group endomorphism
  197. Regular Articles
  198. On pairs of equations in unlike powers of primes and powers of 2
  199. Regular Articles
  200. Differential subordination and convexity criteria of integral operators
  201. Regular Articles
  202. Quantitative relations between short intervals and exceptional sets of cubic Waring-Goldbach problem
  203. Regular Articles
  204. On θ-commutators and the corresponding non-commuting graphs
  205. Regular Articles
  206. Quasi-maximum likelihood estimator of Laplace (1, 1) for GARCH models
  207. Regular Articles
  208. Multiple and sign-changing solutions for discrete Robin boundary value problem with parameter dependence
  209. Regular Articles
  210. Fundamental relation on m-idempotent hyperrings
  211. Regular Articles
  212. A novel recursive method to reconstruct multivariate functions on the unit cube
  213. Regular Articles
  214. Nabla inequalities and permanence for a logistic integrodifferential equation on time scales
  215. Regular Articles
  216. Enumeration of spanning trees in the sequence of Dürer graphs
  217. Regular Articles
  218. Quotient of information matrices in comparison of linear experiments for quadratic estimation
  219. Regular Articles
  220. Fourier series of functions involving higher-order ordered Bell polynomials
  221. Regular Articles
  222. Simple modules over Auslander regular rings
  223. Regular Articles
  224. Weighted multilinear p-adic Hardy operators and commutators
  225. Regular Articles
  226. Guaranteed cost finite-time control of positive switched nonlinear systems with D-perturbation
  227. Regular Articles
  228. A modified quasi-boundary value method for an abstract ill-posed biparabolic problem
  229. Regular Articles
  230. Extended Riemann-Liouville type fractional derivative operator with applications
  231. Topical Issue on Topological and Algebraic Genericity in Infinite Dimensional Spaces
  232. The algebraic size of the family of injective operators
  233. Topical Issue on Topological and Algebraic Genericity in Infinite Dimensional Spaces
  234. The history of a general criterium on spaceability
  235. Topical Issue on Topological and Algebraic Genericity in Infinite Dimensional Spaces
  236. On sequences not enjoying Schur’s property
  237. Topical Issue on Topological and Algebraic Genericity in Infinite Dimensional Spaces
  238. A hierarchy in the family of real surjective functions
  239. Topical Issue on Topological and Algebraic Genericity in Infinite Dimensional Spaces
  240. Dynamics of multivalued linear operators
  241. Topical Issue on Topological and Algebraic Genericity in Infinite Dimensional Spaces
  242. Linear dynamics of semigroups generated by differential operators
  243. Special Issue on Recent Developments in Differential Equations
  244. Isomorphism theorems for some parabolic initial-boundary value problems in Hörmander spaces
  245. Special Issue on Recent Developments in Differential Equations
  246. Determination of a diffusion coefficient in a quasilinear parabolic equation
  247. Special Issue on Recent Developments in Differential Equations
  248. Homogeneous two-point problem for PDE of the second order in time variable and infinite order in spatial variables
  249. Special Issue on Recent Developments in Differential Equations
  250. A nonlinear plate control without linearization
  251. Special Issue on Recent Developments in Differential Equations
  252. Reduction of a Schwartz-type boundary value problem for biharmonic monogenic functions to Fredholm integral equations
  253. Special Issue on Recent Developments in Differential Equations
  254. Inverse problem for a physiologically structured population model with variable-effort harvesting
  255. Special Issue on Recent Developments in Differential Equations
  256. Existence of solutions for delay evolution equations with nonlocal conditions
  257. Special Issue on Recent Developments in Differential Equations
  258. Comments on behaviour of solutions of elliptic quasi-linear problems in a neighbourhood of boundary singularities
  259. Special Issue on Recent Developments in Differential Equations
  260. Coupled fixed point theorems in complete metric spaces endowed with a directed graph and application
  261. Special Issue on Recent Developments in Differential Equations
  262. Existence of entropy solutions for nonlinear elliptic degenerate anisotropic equations
  263. Special Issue on Recent Developments in Differential Equations
  264. Integro-differential systems with variable exponents of nonlinearity
  265. Special Issue on Recent Developments in Differential Equations
  266. Elliptic operators on refined Sobolev scales on vector bundles
  267. Special Issue on Recent Developments in Differential Equations
  268. Multiplicity solutions of a class fractional Schrödinger equations
  269. Special Issue on Recent Developments in Differential Equations
  270. Determining of right-hand side of higher order ultraparabolic equation
  271. Special Issue on Recent Developments in Differential Equations
  272. Asymptotic approximation for the solution to a semi-linear elliptic problem in a thin aneurysm-type domain
  273. Topical Issue on Metaheuristics - Methods and Applications
  274. Learnheuristics: hybridizing metaheuristics with machine learning for optimization with dynamic inputs
  275. Topical Issue on Metaheuristics - Methods and Applications
  276. Nature–inspired metaheuristic algorithms to find near–OGR sequences for WDM channel allocation and their performance comparison
  277. Topical Issue on Cyber-security Mathematics
  278. Monomial codes seen as invariant subspaces
  279. Topical Issue on Cyber-security Mathematics
  280. Expert knowledge and data analysis for detecting advanced persistent threats
  281. Topical Issue on Cyber-security Mathematics
  282. Feedback equivalence of convolutional codes over finite rings
Sign up now to receive a 20% welcome discount
Institutional Access
How does access work?
Have an idea on how to improve our website?
Please write us.
© 2025 De Gruyter Brill
Downloaded on 10.9.2025 from https://www.degruyterbrill.com/document/doi/10.1515/math-2017-0054/html
Scroll to top button