scholarly journals A remark on operating groups

1997 ◽  
Vol 20 (2) ◽  
pp. 393-395
Author(s):  
Yanming Wang
Keyword(s):  
Finite Group ◽  
Short Note ◽  
Invariant Subgroup ◽  
Subnormal Subgroup ◽  
Operator Group ◽  
A Finite Group

LetGbe a finite group andHbe an operator group ofG. In this short note, we show a relationship between subnormal subgroup chains andH-invariant subgroup chains. We remark that the structure ofHis quite restricted whenGhas a specialH-invariant subgroup chain.

Finite Groups with Some Subgroups of Sylow Subgroups s∗-Semipermutable

2021 ◽  
Vol 58 (2) ◽  
pp. 147-156
Author(s):  
Qingjun Kong ◽  
Xiuyun Guo
Keyword(s):  
Finite Group ◽  
Finite Groups ◽  
Sylow Subgroup ◽  
Subnormal Subgroup ◽  
Sylow Subgroups ◽  
A Finite Group

We introduce a new subgroup embedding property in a finite group called s∗-semipermutability. Suppose that G is a finite group and H is a subgroup of G. H is said to be s∗-semipermutable in G if there exists a subnormal subgroup K of G such that G = HK and H ∩ K is s-semipermutable in G. We fix in every non-cyclic Sylow subgroup P of G some subgroup D satisfying 1 < |D| < |P | and study the structure of G under the assumption that every subgroup H of P with |H | = |D| is s∗-semipermutable in G. Some recent results are generalized and unified.

On two sublattices of the subgroup lattice of a finite group

2019 ◽  
Vol 22 (6) ◽  
pp. 1035-1047 ◽  
Author(s):  
Zhang Chi ◽  
Alexander N. Skiba
Keyword(s):  
Finite Group ◽  
Subnormal Subgroup ◽  
Subgroup Lattice ◽  
Chief Factor ◽  
A Finite Group ◽  
Fitting Formation

Abstract Let {\mathfrak{F}} be a non-empty class of groups, let G be a finite group and let {\mathcal{L}(G)} be the lattice of all subgroups of G. A chief {H/K} factor of G is {\mathfrak{F}} -central in G if {(H/K)\rtimes(G/C_{G}(H/K))\in\mathfrak{F}} . Let {\mathcal{L}_{c\mathfrak{F}}(G)} be the set of all subgroups A of G such that every chief factor {H/K} of G between {A_{G}} and {A^{G}} is {\mathfrak{F}} -central in G; {\mathcal{L}_{\mathfrak{F}}(G)} denotes the set of all subgroups A of G with {A^{G}/A_{G}\in\mathfrak{F}} . We prove that the set {\mathcal{L}_{c\mathfrak{F}}(G)} and, in the case when {\mathfrak{F}} is a Fitting formation, the set {\mathcal{L}_{\mathfrak{F}}(G)} are sublattices of the lattice {\mathcal{L}(G)} . We also study conditions under which the lattice {\mathcal{L}_{c\mathfrak{N}}(G)} and the lattice of all subnormal subgroup of G are modular.

On weakly 𝓗-permutable subgroups of finite groups

2019 ◽  
Vol 69 (4) ◽  
pp. 763-772
Author(s):  
Chenchen Cao ◽  
Venus Amjid ◽  
Chi Zhang
Keyword(s):  
Finite Group ◽  
Normal Subgroup ◽  
Finite Groups ◽  
Subnormal Subgroup ◽  
Permutable Subgroups ◽  
A Finite Group ◽  
New Criteria

Abstract Let σ = {σi ∣i ∈ I} be some partition of the set of all primes ℙ, G be a finite group and σ(G) = {σi∣σi ∩ π(G) ≠ ∅}. G is said to be σ-primary if ∣σ(G)∣ ≤ 1. A subgroup H of G is said to be σ-subnormal in G if there exists a subgroup chain H = H0 ≤ H1 ≤ … ≤ Ht = G such that either Hi−1 is normal in Hi or Hi/(Hi−1)Hi is σ-primary for all i = 1, …, t. A set 𝓗 of subgroups of G is said to be a complete Hall σ-set of G if every non-identity member of 𝓗 is a Hall σi-subgroup of G for some i and 𝓗 contains exactly one Hall σi-subgroup of G for every σi ∈ σ(G). Let 𝓗 be a complete Hall σ-set of G. A subgroup H of G is said to be 𝓗-permutable if HA = AH for all A ∈ 𝓗. We say that a subgroup H of G is weakly 𝓗-permutable in G if there exists a σ-subnormal subgroup T of G such that G = HT and H ∩ T ≤ H𝓗, where H𝓗 is the subgroup of H generated by all those subgroups of H which are 𝓗-permutable. By using the weakly 𝓗-permutable subgroups, we establish some new criteria for a group G to be σ-soluble and supersoluble, and we also give the conditions under which a normal subgroup of G is hypercyclically embedded.

scholarly journals Finite groups with given systems of generalised σ-permutable subgroups

2021 ◽  
pp. 25-33
Author(s):  
Viktoria S. Zakrevskaya
Keyword(s):  
Finite Group ◽  
Normal Subgroup ◽  
Finite Groups ◽  
Subnormal Subgroup ◽  
Chief Factor ◽  
Permutable Subgroup ◽  
Permutable Subgroups ◽  
Group A ◽  
A Finite Group

Let σ = {σi|i ∈ I } be a partition of the set of all primes ℙ and G be a finite group. A set ℋ  of subgroups of G is said to be a complete Hall σ-set of G if every member ≠1 of ℋ  is a Hall σi-subgroup of G for some i ∈ I and ℋ contains exactly one Hall σi-subgroup of G for every i such that σi ⌒ π(G)  ≠ ∅.  A group is said to be σ-primary if it is a finite σi-group for some i. A subgroup A of G is said to be: σ-permutable in G if G possesses a complete Hall σ-set ℋ  such that AH x = H  xA for all H ∈ ℋ  and all x ∈ G; σ-subnormal in G if there is a subgroup chain A = A0 ≤ A1 ≤ … ≤ At = G such that either Ai − 1 ⊴ Ai or Ai /(Ai − 1)Ai is σ-primary for all i = 1, …, t; 𝔄-normal in G if every chief factor of G between AG and AG is cyclic. We say that a subgroup H of G is: (i) partially σ-permutable in G if there are a 𝔄-normal subgroup A and a σ-permutable subgroup B of G such that H = < A, B >; (ii) (𝔄, σ)-embedded in G if there are a partially σ-permutable subgroup S and a σ-subnormal subgroup T of G such that G = HT and H ∩ T ≤ S ≤ H. We study G assuming that some subgroups of G are partially σ-permutable or (𝔄, σ)-embedded in G. Some known results are generalised.

On generalized SΦ-supplemented subgroups of finite groups

2019 ◽  
Vol 18 (11) ◽  
pp. 1950204
Author(s):  
Haoran Yu
Keyword(s):  
Finite Group ◽  
Finite Groups ◽  
Subnormal Subgroup ◽  
A Finite Group

According to Mao et al. (2016), let [Formula: see text] be a subgroup of a finite group [Formula: see text]; then [Formula: see text] is said to be generalized [Formula: see text]-supplemented in [Formula: see text] if there exists a subnormal subgroup [Formula: see text] of [Formula: see text] such that [Formula: see text] and [Formula: see text], where [Formula: see text] is the subgroup of [Formula: see text] generated by all those subgroups of [Formula: see text] which are [Formula: see text]-permutable in [Formula: see text]. In this paper, we extend the concept of generalized [Formula: see text]-supplemented subgroups and generalize the main result of Mao et al. (2016).

On ${\mathcal U}_c$-Normal Subgroups of Finite Groups

2007 ◽  
Vol 14 (01) ◽  
pp. 25-36 ◽  
Author(s):  
A. Y. Alsheik Ahmad ◽  
J. J. Jaraden ◽  
Alexander N. Skiba
Keyword(s):  
Finite Group ◽  
Finite Groups ◽  
Subnormal Subgroup ◽  
Normal Subgroups ◽  
Supersoluble Groups ◽  
A Finite Group

Let G be a finite group. We say that a subgroup H of G is [Formula: see text]-normal in G if G has a subnormal subgroup T such that TH = G and (H ∩ T)HG/HG is contained in the [Formula: see text]-hypercenter [Formula: see text] of G/HG, where [Formula: see text] is the class of the finite supersoluble groups. We study the structure of G under the assumption that some subgroups of G are [Formula: see text]-normal in G.

ON SUB-CLASS SIZES OF FINITE GROUPS

2019 ◽  
Vol 108 (3) ◽  
pp. 402-411
Author(s):  
GUOHUA QIAN ◽  
YONG YANG
Keyword(s):  
Finite Group ◽  
Finite Groups ◽  
Subnormal Subgroup ◽  
A Finite Group

For every element $x$ of a finite group $G$, there always exists a unique minimal subnormal subgroup, say, $G_{x}$ of $G$ such that $x\in G_{x}$. The sub-class of $G$ in which $x$ lies is defined by $\{x^{g}\mid g\in G_{x}\}$. The aim of this paper is to investigate the influence of the sub-class sizes on the structure of finite groups.

scholarly journals Finite groups generated by subnormal T-subgroups

1995 ◽  
Vol 37 (3) ◽  
pp. 363-371 ◽  
Author(s):  
John Cossey
Keyword(s):  
Finite Group ◽  
Finite Groups ◽  
Subnormal Subgroup ◽  
A Finite Group

Our aim in this paper is to investigate the restrictions placed on the structure of a finite group if it can be generated by subnormal T-subgroups (a T-group is a group in which every subnormal subgroup is normal). For notational convenience we denote by the class of finite groups that can be generated by subnormal T-subgroups and by the subclass of of those finite groups generated by normal T-subgroups; and for the remainder of this paper we will only consider finite groups.

The sub-class sizes of some elements being square free

2021 ◽  
Author(s):  
Xianhe Zhao ◽  
Yanyan Zhou ◽  
Ruifang Chen ◽  
Qin Huang
Keyword(s):  
Finite Group ◽  
Conjugacy Class ◽  
Finite Groups ◽  
Prime Power ◽  
Prime Factor ◽  
Subnormal Subgroup ◽  
Prime Power Order ◽  
Conjugacy Class Sizes ◽  
A Finite Group

Let [Formula: see text] be an element of a finite group [Formula: see text], and [Formula: see text] a prime factor of the order of [Formula: see text]. It is clear that there always exists a unique minimal subnormal subgroup containing [Formula: see text], say [Formula: see text]. We call the conjugacy class of [Formula: see text] in [Formula: see text] the sub-class of [Formula: see text] in [Formula: see text], see [G. Qian and Y. Yang, On sub-class sizes of finite groups, J. Aust. Math. Soc. (2020) 402–411]. In this paper, assume that [Formula: see text] is the product of the subgroups [Formula: see text] and [Formula: see text], we investigate the solvability, [Formula: see text]-nilpotence and supersolvability of the group [Formula: see text] under the condition that the sub-class sizes of prime power order elements in [Formula: see text] are [Formula: see text] free, [Formula: see text] free and square free, respectively, so that some known results relevant to conjugacy class sizes are generalized.

scholarly journals ON GENERALISED PRONORMAL SUBGROUPS OF FINITE GROUPS

2014 ◽  
Vol 56 (3) ◽  
pp. 691-703 ◽  
Author(s):  
A. BALLESTER-BOLINCHES ◽  
J. C. BEIDLEMAN ◽  
A. D. FELDMAN ◽  
M. F. RAGLAND
Keyword(s):  
Finite Group ◽  
Finite Groups ◽  
Subnormal Subgroup ◽  
Normal Subgroups ◽  
Transitive Relation ◽  
Formula Group ◽  
Permutable Subgroups ◽  
Sylow Permutable ◽  
A Finite Group ◽  
Subnormal Subgroups

AbstractFor a formation $\mathfrak F$, a subgroup M of a finite group G is said to be $\mathfrak F$-pronormal in G if for each g ∈ G, there exists x ∈ 〈U,Ug〉$\mathfrak F$ such that Ux = Ug. Let f be a subgroup embedding functor such that f(G) contains the set of normal subgroups of G and is contained in the set of Sylow-permutable subgroups of G for every finite group G. Given such an f, let fT denote the class of finite groups in which f(G) is the set of subnormal subgroups of G; this is the class of all finite groups G in which to be in f(G) is a transitive relation in G. A subgroup M of a finite group G is said to be $\mathfrak F$-normal in G if G/CoreG(M) belongs to $\mathfrak F$. A subgroup U of a finite group G is called K-$\mathfrak F$-subnormal in G if either U = G or there exist subgroups U = U0 ≤ U1 ≤ . . . ≤ Un = G such that Ui–1 is either normal or $\mathfrak F$-normal in Ui, for i = 1,2, …, n. We call a finite group G an $fT_{\mathfrak F}$-group if every K-$\mathfrak F$-subnormal subgroup of G is in f(G). In this paper, we analyse for certain formations $\mathfrak F$ the structure of $fT_{\mathfrak F}$-groups. We pay special attention to the $\mathfrak F$-pronormal subgroups in this analysis.

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