EUDML

A class ℱ of universal algebras is called a formation if the following conditions are satisfied: 1) Any homomorphic image of A ∈ ℱ is in ℱ; 2) If α₁, α₂ are congruences on A and $A / α_{i} \in ℱ$ , i = 1,2, then A/(α₁∩α₂) ∈ ℱ. We prove that any formation generated by a simple algebra with permutable congruences is minimal, and hence any formation containing a simple algebra, with permutable congruences, contains a minimum subformation. This result gives a partial answer to an open problem of Shemetkov and Skiba...

On $σ$ -permutably embedded subgroups of finite groups

Chenchen Cao; Li Zhang; Wenbin Guo — 2019

Czechoslovak Mathematical Journal

Let $σ = {σ_{i} : i \in I}$ be some partition of the set of all primes $ℙ$ , $G$ be a finite group and $σ (G) = {σ_{i} : σ_{i} \cap π (G) \neq \emptyset}$ . 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 $σ_{i}$ -subgroup of $G$ and $ℋ$ contains exactly one Hall $σ_{i}$ -subgroup of $G$ for every $σ_{i} \in σ (G)$ . $G$ is said to be $σ$ -full if $G$ possesses a complete Hall $σ$ -set. A subgroup $H$ of $G$ is $σ$ -permutable in $G$ if $G$ possesses a complete Hall $σ$ -set $ℋ$ such that $H A^{x}$ = $A^{x} H$ for all $A \in ℋ$ and all $x \in G$ . A subgroup $H$ of $G$ is $σ$ -permutably embedded in $G$ if $H$ is $σ$ -full...

SE-supplemented subgroups of finite groups

Wenbin Guo; Alexander N. Skiba; Nanying Yang — 2013

Rendiconti del Seminario Matematico della Università di Padova

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On a problem of the theory of multiply local formations.

On the influence of the indices of normalizers of Sylow subgroups on the structure of a finite $p$ -soluble group.

On the Shemetkov problem for Fitting classes.

$c$ -semipermutable subgroups of finite groups.

$τ$ -primitive subgroups of finite groups.

Finite groups in which the normalizers of Sylow 3-subgroups are of odd or primary index.

On nonnilpotent groups in which every two 3-maximal subgroups are permutable.

Finite groups in which Sylow normalizers have nilpotent Hall supplements.

Frattini theory for classes of finite universal algebras of Mal'tsev varieties.

$G$ -covering systems of subgroups for classes of $p$ -supersoluble and $p$ -nilpotent finite groups.

$X$ -quasinormal subgroups.

On Hall subgroups of a finite group

Minimal formations of universal algebras

On $σ$ -permutably embedded subgroups of finite groups

SE-supplemented subgroups of finite groups