Laws of the Lattices of $$\sigma $$-Local Formations of Finite Groups

Tsarev, Aleksandr

doi:10.1007/s00009-020-01510-w

Cited by 10 publications

(3 citation statements)

References 35 publications

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“…In computer programming, various abstract data types can be considered as monoids. Because the operation takes two values of a given type and returns a new value of the same type, it can be chained indefinitely, and associativity abstracts away the details of construction; see [40,Example 3.1]. A list (array) is a fine example of a monoid.…”

Section: Inductive Lattices Of Formationsmentioning

confidence: 99%

Algebraic lattices of solvably saturated formations and their applications

Tsarev

Kukharev

2020

Bol. Soc. Mat. Mex.

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In each group G, we select a system of subgroups sðGÞ and say that s is a subgroup functor if G 2 sðGÞ for every group G, and for every epimorphism u : A ! B and any H 2 sðAÞ and T 2 sðBÞ, we have H u 2 sðBÞ and T u À1 2 sðAÞ. We consider only subgroup functors s such that for any group G all subgroups of sðGÞ are subnormal in G. For any set of groups X, the symbol s s ðXÞ denotes the set of groups H such that H 2 sðGÞ for some group G 2 X. A formation F is s-closed if s s ðFÞ ¼ F. The Frattini subgroup UðGÞ of a group G is the intersection of all maximal subgroups of G. A formation F is said to be solvably saturated if it contains each group G with G=UðNÞ 2 F for some solvable normal subgroup N of G. Composition formations are precisely solvably saturated formations. It is shown that the lattice of all s-closed totally composition formations is algebraic. Keywords Finite group Á Subgroup functor Á Formation of groups Á Satellite of formation Á Totally composition formation Á Algebraic lattice of formations Á Formal language Á Hypergroup Mathematics Subject Classification Primary 20F17 Á Secondary 20D10 Á 43A62 Á 20M35 This work has been supported by the Russian Science Foundation under Grant 18-71-10007.

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Section: Inductive Lattices Of Formationsmentioning

confidence: 99%

Algebraic lattices of solvably saturated formations and their applications

Tsarev

Kukharev

2020

Bol. Soc. Mat. Mex.

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“…Corollary 2 [21,Proposition 3.1. ] Let be a -local formation, F be the formation of languages associated with , and let C p be the formation of languages associated with formation of groups where p ∈ ( ) .…”

Section: Example 4 ( -Local Formations)mentioning

confidence: 99%

Classes of monoids with applications: formations of languages and multiply local formations of finite groups

Tsarev

Kukharev

2020

Rend. Circ. Mat. Palermo, II. Ser

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The Frattini subgroup (G) of a group G is the intersection of all maximal subgroups of G. A formation of groups is said to be saturated if G∕ (G) ∈ always implies G ∈ . A formation of finite groups is saturated iff it is local. A local satellite of is a function with domain the set of all primes whose images are formations of finite groups. If the values of this function are themselves local formations, then this circumstance leads to the definition of multiply local formation. The languages corresponding to n-multiply local and totally local formations of finite groups are described.

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“…Skiba [9,10] was proved that the lattice l σ n of all n-multiply σ-local formations of finite groups is algebraic and modular. A.A. Tsarev [16] showed that every law of the lattice of all formations is fulfilled in the lattice l σ n and that for any nonnegative integer n the lattice l σ n is modular but is not distributive. I.N.…”

Section: Introductionmentioning

confidence: 99%

On $τ$-closed $n$-multiply $σ$-local formations of finite groups

Safonova

2021

Preprint

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All groups under consideration are finite. Let σ = {σ i | i ∈ I} be some partition of the set of all primes P, G be a group, and F be a class of groups. Then σ(G)A function f of the form f : σ → {formations of groups} is called a formation σ-function. For any formation σ-function f the class LF σ (f ) is defined as follows:). If for some formation σ-function f we have F = LF σ (f ), then the class F is called σ-local and f is called a σ-local definition of F. Every formation is called 0-multiply σ-local. For n > 0, a formation F is called n-multiply σ-local provided either F = (1) is the class of all identity groups or F = LF σ (f ), where f (σ i ) is (n − 1)-multiply σ-local for all σ i ∈ σ(F). A formation is called totally σ-local if it is n-multiply σ-local for all nonnegative integer n.Let τ (G) be a set of subgroups of G such that G ∈ τ (G). Then τ is called a subgroup functor if for every epimorphism ϕ : A → B and any groups H ∈ τ (A) and T ∈ τ (B) we have H ϕ ∈ τ (B) andIn this paper, we describe some properties of τ -closed n-multiply σ-local formations of finite groups, as well as the main properties of the lattice of such formations. In particular, we prove that the set l τ σn of all τ -closed n-multiply σ-local formations forms a complete modular algebraic lattice of formations. In addition, we proof that the lattice l τ σn is σ-inductive and G-separable.If for some formation σ-function f the equality F = LF σ (f ), then the class F is called σ-local, and f called σ-local definition of F. We write 1) is the class of all identity groups or F = LF σ (f ), where f (σ i ) is (n − 1)-multiply σ-local for all σ i ∈ σ(F). A formation is called totally σ-local if it is n-multiply σ-local for all nonnegative integer n.

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Laws of the Lattices of $$\sigma $$-Local Formations of Finite Groups

Cited by 10 publications

References 35 publications

Algebraic lattices of solvably saturated formations and their applications

Algebraic lattices of solvably saturated formations and their applications

Classes of monoids with applications: formations of languages and multiply local formations of finite groups

On $τ$-closed $n$-multiply $σ$-local formations of finite groups

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