scholarly journals Products of Finite Connected Subgroups

Mathematics ◽  
2020 ◽  
Vol 8 (9) ◽  
pp. 1498
Author(s):  
María Pilar Gállego ◽  
Peter Hauck ◽  
Lev S. Kazarin ◽  
Ana Martínez-Pastor ◽  
María Dolores Pérez-Ramos
Keyword(s):  
Finite Group ◽  
Finite Groups ◽  
Soluble Groups ◽  
Derived Subgroup ◽  
A Finite Group

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 a∈A and b∈B. 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.

Finite groups with two supersoluble subgroups

2019 ◽  
Vol 22 (2) ◽  
pp. 297-312 ◽  
Author(s):  
Victor S. Monakhov ◽  
Alexander A. Trofimuk
Keyword(s):  
Finite Group ◽  
Finite Groups ◽  
Sufficient Conditions ◽  
Maximal Subgroups ◽  
Sylow Subgroups ◽  
Derived Subgroup ◽  
Nilpotent Residual ◽  
A Finite Group

AbstractLetGbe a finite group. In this paper we obtain some sufficient conditions for the supersolubility ofGwith two supersoluble non-conjugate subgroupsHandKof prime index, not necessarily distinct. It is established that the supersoluble residual of such a group coincides with the nilpotent residual of the derived subgroup. We prove thatGis supersoluble in the following cases: one of the subgroupsHorKis nilpotent; the derived subgroup{G^{\prime}}ofGis nilpotent;{|G:H|=q>r=|G:K|}andHis normal inG. Also the supersolubility ofGwith two non-conjugate maximal subgroupsMandVis obtained in the following cases: all Sylow subgroups ofMand ofVare seminormal inG; all maximal subgroups ofMand ofVare seminormal inG.

AN APPROACH TO CAPABLE GROUPS AND SCHUR’S THEOREM

2015 ◽  
Vol 92 (1) ◽  
pp. 52-56
Author(s):  
MITRA HASSANZADEH ◽  
RASOUL HATAMIAN
Keyword(s):  
Finite Group ◽  
Finite Groups ◽  
Finiteness Condition ◽  
Schur’S Theorem ◽  
Derived Subgroup ◽  
Capable Groups ◽  
A Finite Group ◽  
Bounded Rank

Podoski and Szegedy [‘On finite groups whose derived subgroup has bounded rank’, Israel J. Math.178 (2010), 51–60] proved that for a finite group $G$ with rank $r$, the inequality $[G:Z_{2}(G)]\leq |G^{\prime }|^{2r}$ holds. In this paper we omit the finiteness condition on $G$ and show that groups with finite derived subgroup satisfy the same inequality. We also construct an $n$-capable group which is not $(n+1)$-capable for every $n\in \mathbf{N}$.

On residuals of finite groups

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Stefanos Aivazidis ◽  
Thomas Müller
Keyword(s):  
Finite Group ◽  
Finite Groups ◽  
Nilpotent Groups ◽  
Simple Groups ◽  
Fitting Subgroup ◽  
Soluble Groups ◽  
A Finite Group ◽  
Fitting Formation ◽  
Original Theorem

Abstract Theorem C in [S. Dolfi, M. Herzog, G. Kaplan and A. Lev, The size of the solvable residual in finite groups, Groups Geom. Dyn. 1 (2007), 4, 401–407] asserts that, in a finite group with trivial Fitting subgroup, the size of the soluble residual of the group is bounded from below by a certain power of the group order and that the inequality is sharp. Inspired by this result and some of the arguments in the above article, we establish the following generalisation: if 𝔛 is a subgroup-closed Fitting formation of full characteristic which does not contain all finite groups and X ¯ \overline{\mathfrak{X}} is the extension-closure of 𝔛, then there exists an (explicitly known and optimal) constant 𝛾 depending only on 𝔛 such that, for all non-trivial finite groups 𝐺 with trivial 𝔛-radical, | G X ¯ | > | G | γ \lvert G^{\overline{\mathfrak{X}}}\rvert>\lvert G\rvert^{\gamma} , where G X ¯ G^{\overline{\mathfrak{X}}} is the X ¯ \overline{\mathfrak{X}} -residual of 𝐺. When X = N \mathfrak{X}=\mathfrak{N} , the class of finite nilpotent groups, it follows that X ¯ = S \overline{\mathfrak{X}}=\mathfrak{S} , the class of finite soluble groups; thus we recover the original theorem of Dolfi, Herzog, Kaplan, and Lev. In the last section of our paper, building on J. G. Thompson’s classification of minimal simple groups, we exhibit a family of subgroup-closed Fitting formations 𝔛 of full characteristic such that S ⊂ X ¯ ⊂ E \mathfrak{S}\subset\overline{\mathfrak{X}}\subset\mathfrak{E} , where 𝔈 denotes the class of all finite groups, thus providing applications of our main result beyond the reach of the above theorem.

scholarly journals Formations of Π-soluble groups

1969 ◽  
Vol 10 (1-2) ◽  
pp. 241-250 ◽  
Author(s):  
H. Lausch
Keyword(s):  
Finite Group ◽  
Finite Groups ◽  
Soluble Group ◽  
Composition Series ◽  
Soluble Groups ◽  
Hall Subgroups ◽  
A Finite Group

The theory of formations of soluble groups, developed by Gaschütz [4], Carter and Hawkes[1], provides fairly general methods for investigating canonical full conjugate sets of subgroups in finite, soluble groups. Those methods, however, cannot be applied to the class of all finite groups, since strong use was made of the Theorem of Galois on primitive soluble groups. Nevertheless, there is a possiblity to extend the results of the above mentioned papers to the case of Π-soluble groups as defined by Čunihin [2]. A finite group G is called Π-soluble, if, for a given set it of primes, the indices of a composition series of G are either primes belonging to It or they are not divisible by any prime of Π In this paper, we shall frequently use the following result of Čunihin [2]: Ift is a non-empty set of primes, Π′ its complement in the set of all primes, and G is a Π-soluble group, then there always exist Hall Π-subgroups and Hall ′-subgroups, constituting single conjugate sets of subgroups of G respectively, each It-subgroup of G contained in a Hall Π-subgroup of G where each ′-subgroup of G is contained in a Hall Π′-subgroup of G. All groups considered in this paper are assumed to be finite and Π-soluble. A Hall Π-subgroup of a group G will be denoted by G.

scholarly journals SOLVABILITY OF FINITE GROUPS VIA CONDITIONS ON PRODUCTS OF 2-ELEMENTS AND ODD p-ELEMENTS

2010 ◽  
Vol 82 (2) ◽  
pp. 265-273 ◽  
Author(s):  
GIL KAPLAN ◽  
DAN LEVY
Keyword(s):  
Finite Group ◽  
Finite Groups ◽  
London Math ◽  
Characteristic Property ◽  
Soluble Groups ◽  
P Elements ◽  
Odd Order ◽  
Solvability Criterion ◽  
A Finite Group

AbstractWe observe that a solvability criterion for finite groups, conjectured by Miller [The product of two or more groups, Trans. Amer. Math. Soc.12 (1911)] and Hall [A characteristic property of soluble groups, J. London Math. Soc.12 (1937)] and proved by Thompson [Nonsolvable finite groups all of whose local subgroups are solvable, Bull. Amer. Math. Soc.74(3) (1968)], can be sharpened as follows: a finite group is nonsolvable if and only if it has a nontrivial 2-element and an odd p-element, such that the order of their product is not divisible by either 2 or p. We also prove a solvability criterion involving conjugates of odd p-elements. Finally, we define, via a condition on products of p-elements with p′-elements, a formation Pp,p′, for each prime p. We show that P2,2′ (which contains the odd-order groups) is properly contained in the solvable formation.

Finite groups with given systems of σ-semipermutable subgroups

2018 ◽  
Vol 17 (02) ◽  
pp. 1850031 ◽  
Author(s):  
Bin Hu ◽  
Jianhong Huang ◽  
Alexander N. Skiba
Keyword(s):  
Finite Group ◽  
Finite Groups ◽  
Chief Factor ◽  
Transitive Relation ◽  
Soluble Groups ◽  
A Finite Group

Let [Formula: see text] be a partition of the set of all primes [Formula: see text] and [Formula: see text] a finite group. [Formula: see text] is said to be [Formula: see text]-soluble if every chief factor [Formula: see text] of [Formula: see text] is a [Formula: see text]-group for some [Formula: see text]. A set [Formula: see text] of subgroups of [Formula: see text] is said to be a complete Hall [Formula: see text]-set of [Formula: see text] if every member [Formula: see text] of [Formula: see text] is a Hall [Formula: see text]-subgroup of [Formula: see text] for some [Formula: see text] and [Formula: see text] contains exactly one Hall [Formula: see text]-subgroup of [Formula: see text] for every [Formula: see text] such that [Formula: see text]. A subgroup [Formula: see text] of [Formula: see text] is said to be [Formula: see text]-quasinormal or [Formula: see text]-permutable in [Formula: see text] if [Formula: see text] has a complete Hall [Formula: see text]-set [Formula: see text] such that [Formula: see text] for all [Formula: see text] and all [Formula: see text]. We obtain a new characterization of finite [Formula: see text]-soluble groups [Formula: see text] in which [Formula: see text]-permutability is a transitive relation in [Formula: see text].

scholarly journals Finite groups with some weakly pronormal subgroups

2020 ◽  
Vol 18 (1) ◽  
pp. 1742-1747
Author(s):  
Jianjun Liu ◽  
Mengling Jiang ◽  
Guiyun Chen
Keyword(s):  
Finite Group ◽  
Finite Groups ◽  
A Finite Group

Abstract A subgroup H of a finite group G is called weakly pronormal in G if there exists a subgroup K of G such that G = H K G=HK and H ∩ K H\cap K is pronormal in G. In this paper, we investigate the structure of the finite groups in which some subgroups are weakly pronormal. Our results improve and generalize many known results.

scholarly journals A on simplicity criterion for finite groups

1969 ◽  
Vol 10 (3-4) ◽  
pp. 359-362
Author(s):  
Nita Bryce
Keyword(s):  
Finite Group ◽  
Normal Subgroup ◽  
Finite Groups ◽  
Odd Order ◽  
A Finite Group

M. Suzuki [3] has proved the following theorem. Let G be a finite group which has an involution t such that C = CG(t) ≅ SL(2, q) and q odd. Then G has an abelian odd order normal subgroup A such that G = CA and C ∩ A = 〈1〉.

On S-Quasinormal and C-Normal Subgroups of Prime Power Order in Finite Groups

2011 ◽  
Vol 18 (04) ◽  
pp. 685-692
Author(s):  
Xuanli He ◽  
Shirong Li ◽  
Xiaochun Liu
Keyword(s):  
Finite Group ◽  
Finite Groups ◽  
Prime Power ◽  
Maximal Subgroups ◽  
Prime Power Order ◽  
Normal Subgroups ◽  
A Finite Group

Let G be a finite group, p the smallest prime dividing the order of G, and P a Sylow p-subgroup of G with the smallest generator number d. Consider a set [Formula: see text] of maximal subgroups of P such that [Formula: see text]. It is shown that if every member [Formula: see text] of is either S-quasinormally embedded or C-normal in G, then G is p-nilpotent. As its applications, some further results are obtained.

scholarly journals Towards a classification of rigid product quotient varieties of Kodaira dimension 0

2021 ◽  
Author(s):  
Ingrid Bauer ◽  
Christian Gleissner
Keyword(s):  
Finite Group ◽  
Finite Groups ◽  
Structure Theorem ◽  
Complete Classification ◽  
Kodaira Dimension ◽  
The Other ◽  
Diagonal Action ◽  
Quotient Varieties ◽  
A Finite Group

AbstractIn this paper the authors study quotients of the product of elliptic curves by a rigid diagonal action of a finite group G. It is shown that only for $$G = {{\,\mathrm{He}\,}}(3), {\mathbb {Z}}_3^2$$ G = He ( 3 ) , Z 3 2 , and only for dimension $$\ge 4$$ ≥ 4 such an action can be free. A complete classification of the singular quotients in dimension 3 and the smooth quotients in dimension 4 is given. For the other finite groups a strong structure theorem for rigid quotients is proven.

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