soluble group
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Author(s):  
Viachaslau I. Murashka

A Schmidt [Formula: see text]-group is a non-nilpotent [Formula: see text]-group whose proper subgroups are nilpotent and which has the normal Sylow [Formula: see text]-subgroup. The [Formula: see text]-critical graph [Formula: see text] of a finite group [Formula: see text] is a directed graph on the vertex set [Formula: see text] of all prime divisors of [Formula: see text] and [Formula: see text] is an edge of [Formula: see text] if and only if [Formula: see text] has a Schmidt [Formula: see text]-subgroup. The bounds of the nilpotent length of a soluble group are obtained in terms of its [Formula: see text]-critical graph. The structure of a soluble group with given [Formula: see text]-critical graph is obtained in terms of commutators. The connections between [Formula: see text]-critical and other graphs (Sylow, soluble, prime, commuting) of finite groups are found.


Foods ◽  
2021 ◽  
Vol 10 (12) ◽  
pp. 2938
Author(s):  
Marie Bagge Jensen ◽  
Andrius Daugintis ◽  
Jette Jakobsen

Vitamin K is a fat-soluble group of vitamers consisting of phylloquinone (PK) and menaquinones (MKs). To date, only a daily reference intake for PK is set; however, in the last decade, research studying the correlation between MKs intake and improvement of health in regards to cardiovascular diseases, bone metabolism, and chronic kidney disease has been conducted. MKs are synthesised by bacteria in the fermentation process of foods, e.g., cheeses. The content and bioaccessibility of vitamin K vitamers (PK, MK-4, MK-5, MK-6, MK-7, MK-8, MK-9, and MK-10) were assessed in eight different cheese products differing in ripening time, starter culture, fat content, and water content. The bioaccessibility was assessed using the static in vitro digestion model INFOGEST 2.0. Variation of the vitamin K content (<0.5 μg/100 g–32 μg/100 g) and of the vitamin K bioaccessibility (6.4–80%) was observed. A longer ripening time did not necessarily result in an increase of MKs. These results indicate that the vitamin K content and bioaccessibility differs significantly between different cheese products, and the ripening time, starter culture, fat content, and water content cannot explain this difference.


Author(s):  
ELOISA DETOMI ◽  
MARTA MORIGI ◽  
PAVEL SHUMYATSKY

Abstract We show that if w is a multilinear commutator word and G a finite group in which every metanilpotent subgroup generated by w-values is of rank at most r, then the rank of the verbal subgroup $w(G)$ is bounded in terms of r and w only. In the case where G is soluble, we obtain a better result: if G is a finite soluble group in which every nilpotent subgroup generated by w-values is of rank at most r, then the rank of $w(G)$ is at most $r+1$ .


Author(s):  
MARIA FERRARA ◽  
MARCO TROMBETTI

Abstract A subgroup H of a group G is pronormal in G if each of its conjugates $H^g$ in G is conjugate to it in the subgroup $\langle H,H^g\rangle $ ; a group is prohamiltonian if all of its nonabelian subgroups are pronormal. The aim of the paper is to show that a locally soluble group of (regular) cardinality in which all proper uncountable subgroups are prohamiltonian is prohamiltonian. In order to obtain this result, it is proved that the class of prohamiltonian groups is detectable from the behaviour of countable subgroups. Examples are exhibited to show that there are uncountable prohamiltonian groups that do not behave very well. Finally, it is shown that prohamiltonicity can sometimes be detected through the analysis of the finite homomorphic images of a group.


Author(s):  
MATTIA BRESCIA ◽  
ALESSIO RUSSO
Keyword(s):  

Abstract The pronorm of a group G is the set $P(G)$ of all elements $g\in G$ such that X and $X^g$ are conjugate in ${\langle {X,X^g}\rangle }$ for every subgroup X of G. In general the pronorm is not a subgroup, but we give evidence of some classes of groups in which this property holds. We also investigate the structure of a generalised soluble group G whose pronorm contains a subgroup of finite index.


2020 ◽  
Vol 0 (0) ◽  
Author(s):  
Ulderico Dardano ◽  
Fausto De Mari

Abstract We study groups in which each subnormal subgroup is commensurable with a normal subgroup. Recall that two subgroups 𝐻 and 𝐾 are termed commensurable if H ∩ K H\cap K has finite index in both 𝐻 and 𝐾. Among other results, we show that if a (sub)soluble group 𝐺 has the above property, then 𝐺 is finite-by-metabelian, i.e., G ′′ G^{\prime\prime} is finite.


2020 ◽  
Vol 130 (1) ◽  
Author(s):  
Huilong Gu ◽  
Jiao Li ◽  
Huaquan Wei ◽  
Liying Yang
Keyword(s):  

2020 ◽  
Vol 63 (1) ◽  
pp. 121-132
Author(s):  
BIN HU ◽  
JIANHONG HUANG ◽  
ALEXANDER N. SKIBA

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.


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