Let G be a finite group and A, N\leq . Then A_{\operatorname{sn} G} is the subnormal core of A in , that is, subgroup generated by all subgroups contained ; A^{\operatorname{sn} closure intersection containing We say (i) N -subnormal if N\cap G}=N\cap (ii) weakly for some T we have AT=G A\cap T\leq S\leq where S In this paper, consider applications these two concepts. particular, prove soluble only has normal with factor G/N such each maximal chain M_{3}< M_{2}< M_{1}< M_{0}= length...

Throughout this paper, all groups are finite. Let σ={σi|i∈I} be some partition of the set primes P. If n is an integer, symbol σ(n) denotes {σi|σi∩π(n)≠∅}; σ(G)=σ(|G|) and σ(F)=∪G∈Fσ(G). We call any function f form f:σ→{formations groups} a formation σ-function, we put LFσ(f)=(G group|G=1 or G≠1 G/Oσi′,σi(G)∈f(σi) for σi∈σ(G)). σ-function have F=LFσ(f), then say that class F σ-local definition F. suppose every 0-multiply σ-local; > 0, n-multiply provided either F=(1) identity where f(σi)...

It is proved that the lattice of totally saturated formations finite groups modular.

Abstract Let G be a finite group. A subgroup of is said to S-permutable in if permutes with every Sylow P , that is, $AP=PA$ . $A_{sG}$ the generated by all S -permutable subgroups contained and $A^{sG}$ intersection containing We prove soluble group, then -permutability transitive relation only nilpotent residual $G^{\mathfrak {N}}$ avoids pair $(A^{s G}, A_{sG})$ {N}}\cap A^{sG}= G^{\mathfrak A_{sG}$ for subnormal

It is proved that the lattice of totally saturated formations finite groups distributive. Thus, we give an affirmative answer to problem proposed by Shemetkov, Skiba and Guo.

Throughout this article, all groups are finite and G is a group. Let σ={σi|i∈I} be some partition of the set primes P. Then σ(G)={σi|σi∩π(G)≠∅}; σ+(G)={σi|G has chief factor H/K such that σ(H/K)={σi}}. The group said to be: σ-primary if σi-group for i; σ-soluble every σ-primary. symbol Rσ(G) denotes product normal subgroups G. σ-central (in G) (H/K)⋊(G/CG(H/K)) σ-primary; σi-factor σi-group. We say is: σ-nilpotent σ-central; generalized {σi}-nilpotent σ-central. F{gσi}(G) call any function f...

Throughout this paper, all groups are finite. Let $σ=\{σ_i{}|i\in I\}$ be some partition of the set primes $\Bbb{P}$. If $n$ is an integer, $G$ a group, and $\mathfrak{F}$ class groups, then $σ(n)=\{σ_i{}|σ_i{}\cap \pi(n)\ne \emptyset\}$, $σ(G)=σ(|G|)$, $σ(\mathfrak{F})=\cup _G{}_\in{}_\mathfrak{F}σ(G)$. A function $f$ form $f\colon σ\to$ {formations groups} called formation σ-function. For any $σ$-function $LF_σ(f)$ defined as follows: $LF_{\sigma}(f)=(G$ group $|G=1$ или $G\ne1$...

We characterize some classes of finite soluble groups. In particular, we prove that: a group $G$ is supersoluble if and only has normal subgroup $D$ such that $G/D$ avoids every chief factor between $V^{G}$ $V_{G}$ for maximal $V$ the generalized Fitting $F^{*}(G)$ $G$; $PST$-group (that is, Sylow permutability transitive relation on $G$) nilpotent subnormal $A$ $G$.

Let [Formula: see text] be a finite group and the subgroup lattice of text]. A is called: (i) modular in text], if element (in sense Kurosh) text]; (ii) submodular has chain subgroups where for all If then we denote by generated its that are We say text]-modular ([Formula: text]), some containing avoids pair i.e. prove soluble each nilpotent residual modular.

Abstract Let <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>σ</m:mi> <m:mo>=</m:mo> <m:mo stretchy="false">{</m:mo> <m:msub> <m:mi>i</m:mi> </m:msub> fence="true" lspace="0em" rspace="0em">∣</m:mo> <m:mo>∈</m:mo> <m:mi>I</m:mi> </m:mrow> stretchy="false">}</m:mo> </m:math> \sigma=\{\sigma_{i}\mid i\in I\} be some partition of the set all primes and 𝐺 a finite group. Then is said to 𝜎-full if has Hall \sigma_{i} -subgroup for I 𝜎-primary -group 𝑖. In addition, 𝜎-soluble...

We study the properties of lattice c τ ω∞ all -closed totally ω -composition formations finite groups. prove modularity such a for any subgroup functor and nonempty set primes. In particular, we obtain positive answer to question A. N. Skiba L. Shemetkov (2000) about ∞ L formations. establish that is complete sublattice ω- composition