m-embedded Subgroups and p-nilpotency of Finite Groups

2014 ◽  
Vol 57 (4) ◽  
pp. 884-889 ◽  
Author(s):  
Yong Xu ◽  
Xinjian Zhang
Keyword(s):  
Finite Group ◽  
Positive Integer ◽  
Maximal Subgroup ◽  
Finite Groups ◽  
Sufficient Conditions ◽  
Subnormal Subgroup ◽  
A Finite Group

AbstractLet A be a subgroup of a finite group G and ∑ some subgroup series of G. Suppose that for each pair (K,H) such that K is a maximal subgroup of H and Gi-1 ≤ K < H ≤ Gi , for some i, either A ∩ H = A ∩ K or AH = AK. Then A is said to be ∑-embedded in G. And A is said to be m-embedded in G if G has a subnormal subgroup T and a {1 ≤ G}-embedded subgroup C in G such that G = AT and T∩A ≤ C ≤ A. In this article, some sufficient conditions for a finite group G to be p-nilpotent are given whenever all subgroups with order pk of a Sylow p-subgroup of G are m-embedded for a given positive integer k.

On Prime Spectrum of Maximal Subgroups in Finite Groups

2018 ◽  
Vol 25 (04) ◽  
pp. 579-584
Author(s):  
Chi Zhang ◽  
Wenbin Guo ◽  
Natalia V. Maslova ◽  
Danila O. Revin
Keyword(s):  
Finite Group ◽  
Positive Integer ◽  
Maximal Subgroup ◽  
Finite Groups ◽  
Maximal Subgroups ◽  
Prime Spectrum ◽  
Prime Divisors ◽  
A Finite Group ◽  
Theoretical Results

For a positive integer n, we denote by π(n) the set of all prime divisors of n. For a finite group G, the set [Formula: see text] is called the prime spectrum of G. Let [Formula: see text] mean that M is a maximal subgroup of G. We put [Formula: see text] and [Formula: see text]. In this notice, using well-known number-theoretical results, we present a number of examples to show that both K(G) and k(G) are unbounded in general. This implies that the problem “Are k(G) and K(G) bounded by some constant k?”, raised by Monakhov and Skiba in 2016, is solved in the negative.

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.

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.

The Influence of the Number of Non-(sub)normal Non-abelian Subgroups on Solvability of Finite Groups

2016 ◽  
Vol 23 (02) ◽  
pp. 325-328
Author(s):  
Jiangtao Shi ◽  
Cui Zhang
Keyword(s):  
Finite Group ◽  
Finite Groups ◽  
Sufficient Conditions ◽  
Abelian Subgroups ◽  
A Finite Group

We obtain some sufficient conditions on the number of non-(sub)normal non-abelian subgroups of a finite group to be solvable, which extend a result of Shi and Zhang in 2011.

scholarly journals Groups whose Chermak–Delgado lattice is a quasi-antichain

2019 ◽  
Vol 22 (3) ◽  
pp. 529-544
Author(s):  
Lijian An
Keyword(s):  
Finite Group ◽  
Positive Integer ◽  
Finite Groups ◽  
A Finite Group

Abstract A quasi-antichain is a lattice consisting of a maximum, a minimum, and the atoms of the lattice. The width of a quasi-antichain is the number of atoms. For a positive integer {w\geq 3} , a quasi-antichain of width w is denoted by {\mathcal{M}_{w}} . In [B. Brewster, P. Hauck and E. Wilcox, Quasi-antichain Chermak–Delgado lattice of finite groups, Arch. Math. 103 2014, 4, 301–311], it is proved that {\mathcal{M}_{w}} can be the Chermak–Delgado lattice of a finite group if and only if {w=1+p^{a}} for some positive integer a and some prime p. Let t be the number of abelian atoms in {\mathcal{CD}(G)} . In this paper, we completely answer the following question: which values of t are possible in quasi-antichain Chermak–Delgado lattices?

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.

Solubility theorems for finite groups

1973 ◽  
Vol 73 (1) ◽  
pp. 1-6 ◽  
Author(s):  
Thomas J. Laffey
Keyword(s):  
Finite Group ◽  
Finite Groups ◽  
Sufficient Conditions ◽  
A Finite Group

In this paper we obtain various sufficient conditions for the solubility of a finite group. In particular, we show that if G is a finite group and p≥5 is a prime such that all p′-subgroups of G are nilpotent, then G is soluble. We show also that if G is a finite group which has a cyclic Sylow p-subgroup Pand such that for all p′-subgroups H of G, H is nilpotent and H′ is cyclic, then, if p≠3, either P◃G or G has a normal p-complement.

scholarly journals ON WEAKLY -SUPPLEMENTED SUBGROUPS OF SYLOW p-SUBGROUPS OF FINITE GROUPS

2011 ◽  
Vol 53 (2) ◽  
pp. 401-410 ◽  
Author(s):  
LONG MIAO
Keyword(s):  
Finite Group ◽  
Normal Subgroup ◽  
Maximal Subgroup ◽  
Finite Groups ◽  
Prime Divisor ◽  
Proper Subgroup ◽  
A Finite Group

AbstractA subgroup H is called weakly -supplemented in a finite group G if there exists a subgroup B of G provided that (1) G = HB, and (2) if H1/HG is a maximal subgroup of H/HG, then H1B = BH1 < G, where HG is the largest normal subgroup of G contained in H. In this paper we will prove the following: Let G be a finite group and P be a Sylow p-subgroup of G, where p is the smallest prime divisor of |G|. Suppose that P has a non-trivial proper subgroup D such that all subgroups E of P with order |D| and 2|D| (if P is a non-abelian 2-group, |P : D| > 2 and there exists D1 ⊴ E ≤ P with 2|D1| = |D| and E/D1 is cyclic of order 4) have p-nilpotent supplement or weak -supplement in G, then G is p-nilpotent.

On the Frobenius spectrum of a finite group

2017 ◽  
Vol 16 (03) ◽  
pp. 1750051 ◽  
Author(s):  
Jiangtao Shi ◽  
Wei Meng ◽  
Cui Zhang
Keyword(s):  
Finite Group ◽  
Positive Integer ◽  
Finite Groups ◽  
Prime Divisor ◽  
Complete Classification ◽  
Composition Factor ◽  
Frobenius Theorem ◽  
Even Order ◽  
A Finite Group

Let [Formula: see text] be a finite group and [Formula: see text] any divisor of [Formula: see text], the order of [Formula: see text]. Let [Formula: see text], Frobenius’ theorem states that [Formula: see text] for some positive integer [Formula: see text]. We call [Formula: see text] a Frobenius quotient of [Formula: see text] for [Formula: see text]. Let [Formula: see text] be the set of all Frobenius quotients of [Formula: see text], we call [Formula: see text] the Frobenius spectrum of [Formula: see text]. In this paper, we give a complete classification of finite groups [Formula: see text] with [Formula: see text] for [Formula: see text] being the smallest prime divisor of [Formula: see text]. Moreover, let [Formula: see text] be a finite group of even order, [Formula: see text] the set of all Frobenius quotients of [Formula: see text] for even divisors of [Formula: see text] and [Formula: see text] the maximum Frobenius quotient in [Formula: see text], we prove that [Formula: see text] is always solvable if [Formula: see text] or [Formula: see text] and [Formula: see text] is not a composition factor of [Formula: see text].

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.

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