The just non-(virtually abelian) groups with non-trivial Fitting subgroup are classified. Particular attention is given to those which are virtually nilpotent and examples are given of the interesting phenomena that can occur.

Suppose that π be a set of primes and F a local Fitting class. Let Kπ(F) be the set of finite π-soluble groups with a Hall πsubgroup belonging to F. In this paper, we show that the class Kπ(F) is a local Fitting class. Thus, an interesting Shemetkov question for Fitting classes will be answered positively. By using the result, the F-radical of a Hall π-subgroup of a finite π-soluble group is de...

Algorithms are constructed which, when an explicit presentation of a finitely generated metabelian group G in the variety si1 is given, produce unitary presentations for the derived subgroup G', the centre Z{G), the Fitting subgroup Fit(G), and the Frattini subgroup <p(G). Additional algorithms of independent interest are developed for commutative algebra which construct the associated set of p...

We give an example of a pure group that does not have the independence property, whose Fitting subgroup is neither nilpotent nor definable andwhose soluble radical is neither soluble nor definable. This answers a question asked by E. Jaligot in May 2013. The Fitting subgroupof a stable group is nilpotent and definable (F.Wagner [11]). More generally, the Fitting subgroup of a group that satisfi...

Background. Dietary patterns play important role in Helicobacter pylori (H. pylori) infection. We aimed to investigate the potential relationship between Inflammation Index (DII) and H. infection US adults. Methods. This cross-sectional study was based on National Health Nutrition Examination Survey (1999–2000). Individuals aged ≥20 years who provided a 24 hr dietary intake history underwent te...

Abstract We prove that there exists a universal constant D such if p is prime divisor of the index Fitting subgroup finite group G , then number conjugacy classes at least $Dp/\log_2p$ . conjecture we can take $D=1$ and for solvable groups, $D=1/3$

a subgroup $h$ is said to be $nc$-supplemented in a group $g$ if there exists a subgroup $kleq g$ such that $hklhd g$ and $hcap k$ is contained in $h_{g}$, the core of $h$ in $g$. we characterize the supersolubility of finite groups $g$ with that every maximal subgroup of the sylow subgroups is $nc$-supplemented in $g$.

suppose that $h$ is a subgroup of $g$‎, ‎then $h$ is said to be‎ ‎$s$-permutable in $g$‎, ‎if $h$ permutes with every sylow subgroup of‎ ‎$g$‎. ‎if $hp=ph$ hold for every sylow subgroup $p$ of $g$ with $(|p|‎, ‎|h|)=1$)‎, ‎then $h$ is called an $s$-semipermutable subgroup of $g$‎. ‎in this paper‎, ‎we say that $h$ is partially $s$-embedded in $g$ if‎ ‎$g$ has a normal subgroup $t$ such that $ht...