S-semipermutability of subgroups of p-nilpotent residual and p-supersolubility of a finite group

2020 ◽  
pp. 2150117
Author(s):  
Zhengtian Qiu ◽  
Shouhong Qiao
Keyword(s):  
Finite Group ◽  
Nilpotent Residual ◽  
A Finite Group

A subgroup [Formula: see text] of a finite group [Formula: see text] is said to be [Formula: see text]-semipermutable in [Formula: see text], if [Formula: see text] permutes with all Sylow [Formula: see text]-subgroups of [Formula: see text] for the primes [Formula: see text] not dividing [Formula: see text]. In this paper, we consider the [Formula: see text]-semipermutability of some [Formula: see text]-subgroups to investigate the [Formula: see text]-supersolubility of a finite group. Some interesting results are obtained which extend some known results.

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.

BOUNDING THE ORDER OF THE NILPOTENT RESIDUAL OF A FINITE GROUP

2016 ◽  
Vol 94 (2) ◽  
pp. 273-277
Author(s):  
AGENOR FREITAS DE ANDRADE ◽  
PAVEL SHUMYATSKY
Keyword(s):  
Finite Group ◽  
Positive Integer ◽  
Lower Central Series ◽  
Central Series ◽  
Nilpotent Residual ◽  
A Finite Group

The last term of the lower central series of a finite group $G$ is called the nilpotent residual. It is usually denoted by $\unicode[STIX]{x1D6FE}_{\infty }(G)$. The lower Fitting series of $G$ is defined by $D_{0}(G)=G$ and $D_{i+1}(G)=\unicode[STIX]{x1D6FE}_{\infty }(D_{i}(G))$ for $i=0,1,2,\ldots \,$. These subgroups are generated by so-called coprime commutators $\unicode[STIX]{x1D6FE}_{k}^{\ast }$ and $\unicode[STIX]{x1D6FF}_{k}^{\ast }$ in elements of $G$. More precisely, the set of coprime commutators $\unicode[STIX]{x1D6FE}_{k}^{\ast }$ generates $\unicode[STIX]{x1D6FE}_{\infty }(G)$ whenever $k\geq 2$ while the set $\unicode[STIX]{x1D6FF}_{k}^{\ast }$ generates $D_{k}(G)$ for $k\geq 0$. The main result of this article is the following theorem: let $m$ be a positive integer and $G$ a finite group. Let $X\subset G$ be either the set of all $\unicode[STIX]{x1D6FE}_{k}^{\ast }$-commutators for some fixed $k\geq 2$ or the set of all $\unicode[STIX]{x1D6FF}_{k}^{\ast }$-commutators for some fixed $k\geq 1$. Suppose that the size of $a^{X}$ is at most $m$ for any $a\in G$. Then the order of $\langle X\rangle$ is $(k,m)$-bounded.

scholarly journals Almost Engel finite and profinite groups

2016 ◽  
Vol 26 (05) ◽  
pp. 973-983 ◽  
Author(s):  
E. I. Khukhro ◽  
P. Shumyatsky
Keyword(s):  
Finite Group ◽  
Normal Subgroup ◽  
Positive Integer ◽  
Profinite Group ◽  
Profinite Groups ◽  
Locally Nilpotent ◽  
Nilpotent Residual ◽  
A Finite Group

Let [Formula: see text] be an element of a group [Formula: see text]. For a positive integer [Formula: see text], let [Formula: see text] be the subgroup generated by all commutators [Formula: see text] over [Formula: see text], where [Formula: see text] is repeated [Formula: see text] times. We prove that if [Formula: see text] is a profinite group such that for every [Formula: see text] there is [Formula: see text] such that [Formula: see text] is finite, then [Formula: see text] has a finite normal subgroup [Formula: see text] such that [Formula: see text] is locally nilpotent. The proof uses the Wilson–Zelmanov theorem saying that Engel profinite groups are locally nilpotent. In the case of a finite group [Formula: see text], we prove that if, for some [Formula: see text], [Formula: see text] for all [Formula: see text], then the order of the nilpotent residual [Formula: see text] is bounded in terms of [Formula: see text].

Splitting over nilpotent and hypercentral residuals

1975 ◽  
Vol 78 (2) ◽  
pp. 215-226 ◽  
Author(s):  
B. Hartley ◽  
M. J. Tomkinson
Keyword(s):  
Finite Group ◽  
Finite Groups ◽  
Formation Theory ◽  
Locally Finite ◽  
Locally Finite Groups ◽  
Nilpotent Residual ◽  
A Finite Group

It is a well known theorem of Gaschütz (4) and Schenkman (12) that if G is a finite group whose nilpotent residual A is Abelian, then G splits over A and the complements to A in G are conjugate. Following Robinson (10) we describe this situation by saying that G splits conjugately over A. A number of generalizations of this result have since been obtained, some of them being in the context of the formation theory of finite or locally finite groups (see, for example, (1), (3)) and others, for example, the recent and far-reaching results of Robinson (10, 11) being concerned with groups which are not necessarily periodic. Our results here are of the latter type.

A sufficient condition for nilpotency of the nilpotent residual of a finite group

2018 ◽  
Vol 21 (2) ◽  
pp. 289-293 ◽  
Author(s):  
Agenor Freitas de Andrade ◽  
Alex Carrazedo Dantas
Keyword(s):  
Finite Group ◽  
Sufficient Condition ◽  
Nilpotent Residual ◽  
A Finite Group

AbstractLetGbe a finite group with the property that if{a,b}are powers of{\delta_{1}^{*}}-commutators such that{(|a|,|b|)=1}, then{|ab|=|a||b|}. We show that{\gamma_{\infty}(G)}is nilpotent.

ON THE σ-NILPOTENT NORM AND THE σ-NILPOTENT LENGTH OF A FINITE GROUP

2020 ◽  
Vol 63 (1) ◽  
pp. 121-132
Author(s):  
BIN HU ◽  
JIANHONG HUANG ◽  
ALEXANDER N. SKIBA
Keyword(s):  
Finite Group ◽  
Soluble Group ◽  
Normal Subgroups ◽  
Nilpotent Length ◽  
Nilpotent Residual ◽  
A Finite Group ◽  
Primary Groups ◽  
Relationship Of ◽  
The Relationship ◽  
Sigma G

AbstractLet G be a finite group and σ = {σi| i ∈ I} some partition of the set of all primes $\Bbb{P}$ . Then G is said to be: σ-primary if G is a σi-group for some i; σ-nilpotent if G = G1× … × Gt for some σ-primary groups G1, … , Gt; σ-soluble if every chief factor of G is σ-primary. We use $G^{{\mathfrak{N}}_{\sigma}}$ to denote the σ-nilpotent residual of G, that is, the intersection of all normal subgroups N of G with σ-nilpotent quotient G/N. If G is σ-soluble, then the σ-nilpotent length (denoted by lσ (G)) of G is the length of the shortest normal chain of G with σ-nilpotent factors. Let Nσ (G) be the intersection of the normalizers of the σ-nilpotent residuals of all subgroups of G, that is, $${N_\sigma }(G) = \bigcap\limits_{H \le G} {{N_G}} ({H^{{_\sigma }}}).$$ Then the subgroup Nσ (G) is called the σ-nilpotent norm of G. We study the relationship of the σ-nilpotent length with the σ-nilpotent norm of G. In particular, we prove that the σ-nilpotent length of a σ-soluble group G is at most r (r > 1) if and only if lσ (G/ Nσ (G)) ≤ r.

Subnormality and residuals for saturated formations: A generalization of Schenkman’s theorem

2020 ◽  
Vol 0 (0) ◽  
Author(s):  
Stefanos Aivazidis ◽  
Inna N. Safonova ◽  
Alexander N. Skiba
Keyword(s):  
Finite Group ◽  
Nilpotent Groups ◽  
Subnormal Subgroup ◽  
Saturated Formation ◽  
Chief Factor ◽  
Normal Subgroups ◽  
Hereditary Saturated Formation ◽  
Nilpotent Residual ◽  
Saturated Formations ◽  
A Finite Group

Abstract Let 𝐺 be a finite group, and let 𝔉 be a hereditary saturated formation. We denote by Z F ⁢ ( G ) \mathbf{Z}_{\mathfrak{F}}(G) the product of all normal subgroups 𝑁 of 𝐺 such that every chief factor H / K H/K of 𝐺 below 𝑁 is 𝔉-central in 𝐺, that is, ( H / K ) ⋊ ( G / C G ⁢ ( H / K ) ) ∈ F (H/K)\rtimes(G/\mathbf{C}_{G}(H/K))\in\mathfrak{F} . A subgroup A ⩽ G A\leqslant G is said to be 𝔉-subnormal in the sense of Kegel, or 𝐾-𝔉-subnormal in 𝐺, if there is a subgroup chain A = A 0 ⩽ A 1 ⩽ ⋯ ⩽ A n = G A=A_{0}\leqslant A_{1}\leqslant\cdots\leqslant A_{n}=G such that either A i - 1 ⁢ ⊴ ⁢ A i A_{i-1}\trianglelefteq A_{i} or A i / ( A i - 1 ) A i ∈ F A_{i}/(A_{i-1})_{A_{i}}\in\mathfrak{F} for all i = 1 , … , n i=1,\ldots,n . In this paper, we prove the following generalization of Schenkman’s theorem on the centraliser of the nilpotent residual of a subnormal subgroup: Let 𝔉 be a hereditary saturated formation containing all nilpotent groups, and let 𝑆 be a 𝐾-𝔉-subnormal subgroup of 𝐺. If Z F ⁢ ( E ) = 1 \mathbf{Z}_{\mathfrak{F}}(E)=1 for every subgroup 𝐸 of 𝐺 such that S ⩽ E S\leqslant E , then C G ⁢ ( D ) ⩽ D \mathbf{C}_{G}(D)\leqslant D , where D = S F D=S^{\mathfrak{F}} is the 𝔉-residual of 𝑆.

ON HIGHER FROBENIUS–SCHUR INDICATORS

2021 ◽  
pp. 1-11
Author(s):  
YANJUN LIU ◽  
WOLFGANG WILLEMS
Keyword(s):  
Finite Group ◽  
Representation Theory ◽  
Irreducible Characters ◽  
A Finite Group

Abstract Similarly to the Frobenius–Schur indicator of irreducible characters, we consider higher Frobenius–Schur indicators $\nu _{p^n}(\chi ) = |G|^{-1} \sum _{g \in G} \chi (g^{p^n})$ for primes p and $n \in \mathbb {N}$ , where G is a finite group and $\chi $ is a generalised character of G. These invariants give answers to interesting questions in representation theory. In particular, we give several characterisations of groups via higher Frobenius–Schur indicators.

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