Paper «On finite p-groups of supersoluble type» published in J. Algebra

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A. Ballester-Bolinches, R. Esteban-Romero, H. Meng, and N. Su.
On finite p-groups of supersoluble type.
J. Algebra, 567:1–10, 2021.

doi:10.1016/j.jalgebra.2020.08.025

Abstract

A finite p-group S is said to be of supersoluble type if every fusion system over S is supersoluble. The main aim of this paper is to characterise the finite p-groups of supersoluble type. Abelian and metacyclic p-groups of supersoluble type are completely described. Furthermore, we show that the Sylow p-subgroups of supersoluble type of a finite simple group must be cyclic.

2020 Mathematics Subject Classification: 20D20; 20D15; 20D05

Keywords: finite group; fusion system; supersolubility

Paper «Products of groups and class sizes of π-elements» published in Mediterr. J. Math.

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M. J. Felipe, A. Martínez-Pastor, V. M. Ortiz-Sotomayor.
Products of groups and class sizes of π-elements.
Mediterr. J. Math., 17(1):Paper No. 15, 20, 2020.

doi:10.1007/s00009-019-1444-5

Abstract

We provide structural criteria for some finite factorised groups G=AB when the conjugacy class sizes in G of certain π-elements in AB are either π-numbers or π′-numbers, for a set of primes π. In particular, we extend for products of groups some earlier results.

2020 Mathematics Subject Classification: 20D10, 20D40, 20E45, 20D20

Keywords: finite group; products of groups; conjugacy classes, π-structure

Paper «On the σ-Length of Maximal Subgroups of Finite σ-Soluble Groups» published in Mathematics

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Abd El-Rahman Heliel, Mohammed Al-Shomrani, Adolfo Ballester-Bolinches.
On the σ-Length of Maximal Subgroups of Finite σ-Soluble Groups.
Mathematics, 8(12):2165 (4 pages), 2020.

doi:10.3390/math8122165

Abstract

Let σ={σi:iI} be a partition of the set P of all prime numbers and let G be a finite group. We say that G is σ-primary if all the prime factors of |G| belong to the same member of σ. G is said to be σ-soluble if every chief factor of G is σ-primary, and G is σ-nilpotent if it is a direct product of σ-primary groups. It is known that G has a largest normal σ-nilpotent subgroup which is denoted by (G). Let n be a non-negative integer. The n-term of the σ-Fitting series of G is defined inductively by F0(G)=1, and Fn+1(G)/Fn(G)=(G/Fn(G)). If G is σ-soluble, there exists a smallest n such that Fn(G)=G. This number n is called the σ-nilpotent length of G and it is denoted by (G). If F is a subgroup-closed saturated formation, we define the σ-F-length (G,F) of G as the σ-nilpotent length of the F-residual GF of G. The main result of the paper shows that if A is a maximal subgroup of G and G is a σ-soluble, then (A,F)=(G,F)−i for some i∈{0,1,2}.

Keywords: finite group; σ-solubility; σ-nilpotency; σ-nilpotent length

Paper «The Dπ-property on products of π-decomposable groups» published in Rev. R. Acad. Cienc. Exactas Fís. Nat. Ser. A Mat. RACSAM

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L. S. Kazarin, A. Martínez-Pastor, and M. D. Pérez-Ramos.
The Dπ-property on products of π-decomposable groups.
Rev. R. Acad. Cienc. Exactas Fís. Nat. Ser. A Mat. RACSAM, 115(1):Paper No. 13, 18, 2021.

doi:10.1007/s13398-020-00950-z

Abstract

The aim of this paper is to prove the following result: Let π be a set of odd primes. If the group G = AB is the product of two π-decomposable subgroups A = Aπ × Aπ′ and B = Bπ × Bπ′, then G has a unique conjugacy class of Hall π-subgroups, and any π-subgroup is contained in a Hall π-subgroup (i.e. G satisfies property Dπ).

2020 Mathematics Subject Classification: 20D40; 20D20; 20E32

Keywords: finite groups; product of subgroups; π-structure; simple groups

Paper «On σ-subnormal subgroups of factorised finite groups» published in J. Algebra

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A. Ballester-Bolinches, S. F. Kamornikov, M. C. Pedraza-Aguilera, and X. Yi.
On σ-subnormal subgroups of factorised finite groups.
J. Algebra, 559:195–202, 2020.

doi:10.1016/j.jalgebra.2020.05.002

Abstract

Let σ = {σi : iI} be a partition of the set ℙ of all prime numbers. A subgroup X of a finite group G is called σsubnormal in G if there is chain of subgroups X = X0X1 ⊆⋯⊆ Xn = G with Xj-1 normal in Xj or Xi/CoreXi(Xi-1) is a σ-group for some iI, 1 ≤ jn. In the special case that σ is the partition of ℙ into sets containing exactly one prime each, the σ-subnormality reduces to the familiar case of subnormality.

If a finite soluble group G = AB is factorised as the product of the subgroups A and B, and X is a subgroup of G such that X is σ-subnormal in 〈X, Xg〉 for all gAB , we prove that X is σ-subnormal in G. This is an extension of a subnormality criteria due to Maier and Sidki and Casolo.

2020 Mathematics Subject Classification: 20D10, 20D20

Keywords: Finite group; Soluble group; σ-Subnormal subgroup; σ-Nilpotency; Factorised group

Paper «Products of finite connected subgroups» published in Mathematics

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María Pilar Gállego, Peter Hauck, Lev S. Kazarin, Ana Martínez-Pastor, and María Dolores Pérez-Ramos.
Products of finite connected subgroups.
Mathematics, 18(9):1498 (8 pages), 2020.

doi:10.3390/math8091498

Abstract

For a non-empty class of groups L, a finite group G=AB is said to be an L-connected product of the subgroups A and B if ⟨a,b⟩∈L for all aA and bB. In a previous paper, we prove that, for such a product, when L=S is the class of finite soluble groups, then [A,B] is soluble. This generalizes the theorem of Thompson that states the solubility of finite groups whose two-generated subgroups are soluble. In the present paper, our result is applied to extend to finite groups previous research about finite groups in the soluble universe. In particular, we characterize connected products for relevant classes of groups, among others, the class of metanilpotent groups and the class of groups with nilpotent derived subgroup. Additionally, we give local descriptions of relevant subgroups of finite groups.

2020 Mathematics Subject Classification: 20D40, 20E45, 20D20, 20D60

Keywords: finite groups; products of subgroups; two-generated subgroups; L-connection; Fitting classes; Fitting series; formations

Paper «On σ-subnormal closure» published in Comm. Algebra

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M. M. Al-Shomrani, A. A. Heliel, and Adolfo Ballester-Bolinches.
On σ-subnormal closure.
Comm. Algebra, 48(8):3624–3627, 2020.

doi:10.1080/00927872.2020.1742348

Abstract

Let σ = {σi : iI} be a partition of the set ℙ of all prime numbers. A subgroup A of a finite group G is called σsubnormal in G if there is chain of subgroups A = A0A1 ⊆⋯⊆ An = G with Aj-1 normal in Aj or Ai/CoreAi(Ai-1) is a σj-group for some jI, 1 ≤ in. In this paper, the description of the unique smallest σ-subnormal subgroup of a σ-soluble group containing a given subgroup is obtained.

2020 Mathematics Subject Classification: 20D10, 20D20

Keywords: Finite group; σ-soluble group; σ-subnormal subgroup

Paper «On products of groups and indices not divisible by a given prime» published in Monatsh. Math.

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María José Felipe, Lev S. Kazarin, Ana Martínez-Pastor, and Víctor Sotomayor.
On products of groups and indices not divisible by a given prime.
Comm. Algebra, 193(4):811–827, 2020.

doi:10.1007/s00605-020-01446-z

Abstract

Let the group G = AB be the product of subgroups A and B, and let p be a prime. We prove that p does not divide the conjugacy class size (index) of each p-regular element of prime power order xAB if and only if G is p-decomposable, i.e. G=Op(G) × Op’(G).

2020 Mathematics Subject Classification: 20D40, 20E45, 20D20, 20D60

Keywords: Finite groups; products of groups; conjugacy classes; p-structure; prime graph; almost simple groups

Paper «On large orbits of actions of finite soluble groups: applications» published in Recent advances in pure and applied mathematics

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Adolfo Ballester-Bolinches, Ramon Esteban-Romero, and H. Meng.
On large orbits of actions of finite soluble groups: applications.
Recent advances in pure and applied mathematics. Based on contributions presented at the Second Joint Meeting Spain-Brazil in Mathematics, Cádiz, Spain, December 11–14, 2018, pages 105–113. Cham: Springer, 2020.

doi:10.1007/978-3-030-41321-7_8

Abstract

The main aim of this survey paper is to present two orbit theorems and to show how to apply them to obtain a result that can be regarded as a significant step towards the solution of Gluck’s conjecture on large character degrees of finite soluble groups. We also show how to apply them to solve questions about intersections of some conjugacy families of subgroups of finite soluble groups.

2020 Mathematics Subject Classification: 20C15, 20D10, 20D20, 20D45

Keywords: finite groups, soluble groups, linear groups, regular orbits, formations, prefrattini subgroups, system normalisers

Paper «The number of maximal subgroups and probabilistic generation of finite groups» published in Int. J. Group Theory

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Adolfo Ballester-Bolinches, Ramón Esteban-Romero, Paz Jiménez-Seral, Hangyang Meng.
The number of maximal subgroups and probabilistic generation of finite groups.
Int. J. Group Theory, 9(1):31–42, 2020.

doi:10.22108/ijgt.2019.114469.1521

Abstract

In this survey we present some significant bounds for the‎ ‎number of maximal subgroups of a given index of a finite group‎. ‎As a‎ ‎consequence‎, ‎new bounds for the number of random‎ ‎generators needed to generate a finite d-generated group with high‎ ‎probability which are significantly tighter than the ones obtained in‎ ‎the paper of Jaikin-Zapirain and Pyber (Random generation of finite‎ ‎and profinite groups and group enumeration‎, Ann. Math.‎, 183 (2011) 769–814) are obtained‎. ‎The results of‎ ‎Jaikin-Zapirain and Pyber‎, ‎as well as other results of Lubotzky‎, ‎Detomi‎, ‎and Lucchini‎, ‎appear as particular cases of our theorems‎.

2020 Mathematics Subject Classification: 20P05

Keywords: finite group; maximal subgroup; probabilistic generation; primitive group