A generalization of Kramer’s theory

  • Z. ChiEmail author
  • A. N. Skiba


Throughout this paper, all groups are finite. \({\sigma =\{\sigma_{i}\mid i\in I \}}\) is some partition of the set of all primes \({\mathbb{P}}\), and \({\sigma (n)= \{\sigma _{i}\mid \sigma _{i}\cap \pi (n)\ne \emptyset \}}\) for any \({n\in \mathbb{N}}\). The natural numbers n and m are called \({\sigma}\)-coprime if \({\sigma (n)\cap \sigma (m)=\emptyset}\).

Let t >  1 be a natural number and let \({\mathfrak{F}}\) be a class of groups. Then we say that \({\mathfrak{F}}\) is \({\Gamma_{t}^{\sigma}}\)-closed (respectively weakly\({\Gamma_{t}^{\sigma}}\)-closed) provided \({\mathfrak{F}}\) contains each finite group G which satisfies the following conditions:

  1. (1)

    G has subgroups \({A_{1}, \ldots, A_{t} \in \mathfrak{F}}\) such that G = AiAj for all \({i\ne j}\) ;

  2. (2)

    The indices \({|G:N_{G}(A_{1})|, \ldots, |G:N_{G}(A_{t})|}\) (respectively the indices \({|{G:A_{1}}|, \ldots, |G:A_{t-1}|, |G:N_{G}(A_{t})|}\)) are pairwise \({\sigma}\)-coprime.


We study properties and some applications of (weakly) \({\Gamma_{t}^{\sigma}}\)-closed classes of finite groups.

Key words and phrases

finite group formation \({\sigma}\)-function \({\sigma}\)-local formation (weakly) \({\Gamma_{t}^{\sigma}}\)-closed class of groups \({\sigma}\)-soluble group \({\sigma}\)-nilpotent group 

Mathematics Subject Classification

20D10 20D15 20D20 


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Copyright information

© Akadémiai Kiadó, Budapest, Hungary 2019

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of Science and Technology of ChinaHefeiP. R. China
  2. 2.Department of Mathematics and Technologies of ProgrammingFrancisk Skorina Gomel State UniversityGomelBelarus

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