Finite groups in which every non-abelian subgroup is a TI-subgroup or a subnormal subgroup

2019 ◽  
Vol 18 (08) ◽  
pp. 1950159
Author(s):  
Jiangtao Shi
Keyword(s):  
Finite Group ◽  
Finite Groups ◽  
Abelian Subgroup ◽  
Subnormal Subgroup ◽  
Cyclic Subgroups ◽  
Abelian Subgroups ◽  
A Finite Group

It is known that a TI-subgroup of a finite group may not be a subnormal subgroup and a subnormal subgroup of a finite group may also not be a TI-subgroup. For the non-abelian subgroups, we prove that if every non-abelian subgroup of a finite group [Formula: see text] is a TI-subgroup or a subnormal subgroup, then every non-abelian subgroup of [Formula: see text] must be subnormal in [Formula: see text]. We also show that the non-cyclic subgroups have the same property.

Finite groups whose non-abelian self-centralizing subgroups are TI-subgroups or subnormal subgroups

2020 ◽  
pp. 2150040
Author(s):  
Yuqing Sun ◽  
Jiakuan Lu ◽  
Wei Meng
Keyword(s):  
Finite Group ◽  
Finite Groups ◽  
Abelian Subgroup ◽  
Subnormal Subgroup ◽  
A Finite Group ◽  
Subnormal Subgroups

In this paper, we prove that if every non-abelian self-centralizing subgroup of a finite group [Formula: see text] is a TI-subgroup or a subnormal subgroup of [Formula: see text], then every non-abelian subgroup of [Formula: see text] must be subnormal in [Formula: see text].

Finite Groups in Which the Automizers of Some Abelian Subgroups are Trivial

2018 ◽  
Vol 25 (04) ◽  
pp. 701-712
Author(s):  
Pengfei Bai ◽  
Xiuyun Guo
Keyword(s):  
Finite Group ◽  
Finite Groups ◽  
Abelian Subgroup ◽  
Group Of Automorphisms ◽  
Normal Abelian Subgroup ◽  
Abelian Subgroups ◽  
A Finite Group

If H is a subgroup of a finite group G, then the automizer AutG(H) of H in G is defined as the group of automorphisms of H induced by conjugation by elements of NG(H). A finite group G is called an NNC-group if for any non-normal abelian subgroup A, either [Formula: see text] or [Formula: see text]. In this paper, classifications of nilpotent NNC-groups and non-solvable NNC-groups are given. We also investigate the solvable NNC-groups and describe the structure of solvable NNC-groups.

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.

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.

FINITE GROUPS WITH MULTIPLICITY-FREE PERMUTATION CHARACTERS

2005 ◽  
Vol 04 (02) ◽  
pp. 187-194
Author(s):  
MICHITAKU FUMA ◽  
YASUSHI NINOMIYA
Keyword(s):  
Finite Group ◽  
Finite Groups ◽  
Abelian Subgroup ◽  
Hecke Algebra ◽  
Endomorphism Algebra ◽  
Theoretic Method ◽  
A Finite Group ◽  
Multiplicity Free ◽  
Canonical Involution

Let G be a finite group and H a subgroup of G. The Hecke algebra ℋ(G,H) associated with G and H is defined by the endomorphism algebra End ℂ[G]((ℂH)G), where ℂH is the trivial ℂ[H]-module and (ℂH)G = ℂH⊗ℂ[H] ℂ[G]. As is well known, ℋ(G,H) is a semisimple ℂ-algebra and it is commutative if and only if (ℂH)G is multiplicity-free. In [6], by a ring theoretic method, it is shown that if the canonical involution of ℋ(G,H) is the identity then ℋ(G,H) is commutative and, if there exists an abelian subgroup A of G such that G = AH then ℋ(G,H) is commutative. In this paper, by a character theoretic method, we consider the commutativity of ℋ(G,H).

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.

Finite groups with each non-normal subgroup having maximal normal cores

2016 ◽  
Vol 15 (03) ◽  
pp. 1650053
Author(s):  
Heng Lv ◽  
Zhibo Shao ◽  
Wei Zhou
Keyword(s):  
Finite Group ◽  
Normal Subgroup ◽  
Finite Groups ◽  
Abelian Subgroup ◽  
A Finite Group

In this paper, we study a finite group [Formula: see text] such that [Formula: see text] is a prime for each non-normal subgroup [Formula: see text] of [Formula: see text]. We prove that such a group must contain a big abelian subgroup. More specifically, if such a group [Formula: see text] is not supersoluble, then there is an abelian subgroup [Formula: see text] such that [Formula: see text], and if [Formula: see text] is supersoluble, then there is an abelian subgroup [Formula: see text] such that [Formula: see text] or [Formula: see text], where [Formula: see text] and [Formula: see text] are primes.

The Influence of Certain Abelian Subgroups on the Structure of Finite Groups

2008 ◽  
Vol 15 (03) ◽  
pp. 479-484 ◽  
Author(s):  
M. Ramadan
Keyword(s):  
Finite Group ◽  
Finite Groups ◽  
Prime Power ◽  
Prime Power Order ◽  
Group A ◽  
Abelian Subgroups ◽  
A Finite Group

Let G be a finite group. A subgroup K of a group G is called an [Formula: see text]-subgroup of G if NG(K) ∩ Kx ≤ K for all x ∈ G. The set of all [Formula: see text]-subgroups of G is denoted by [Formula: see text]. In this paper, we investigate the structure of a group G under the assumption that certain abelian subgroups of prime power order belong to [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.

THE SECOND MINIMUM/MAXIMUM VALUE OF THE NUMBER OF CYCLIC SUBGROUPS OF FINITE -GROUPS

2020 ◽  
pp. 1-8
Author(s):  
MIHAI-SILVIU LAZOREC ◽  
RULIN SHEN ◽  
MARIUS TĂRNĂUCEANU
Keyword(s):  
Finite Group ◽  
Finite Groups ◽  
Cyclic Subgroups ◽  
Minimum Value ◽  
Maximum Value ◽  
A Finite Group ◽  
Minimum Points

Let $C(G)$ be the poset of cyclic subgroups of a finite group $G$ and let $\mathscr{P}$ be the class of $p$ -groups of order  $p^{n}$ ( $n\geq 3$ ). Consider the function $\unicode[STIX]{x1D6FC}:\mathscr{P}\longrightarrow (0,1]$ given by $\unicode[STIX]{x1D6FC}(G)=|C(G)|/|G|$ . In this paper, we determine the second minimum value of  $\unicode[STIX]{x1D6FC}$ , as well as the corresponding minimum points. Since the problem of finding the second maximum value of $\unicode[STIX]{x1D6FC}$ has been solved for $p=2$ , we focus on the case of odd primes in determining the second maximum.

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