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A class of commutative loops with metacyclic inner mapping groups

Aleš Drápal (2008)

Commentationes Mathematicae Universitatis Carolinae

We investigate loops defined upon the product ℤ m × ℤ k by the formula ( a , i ) ( b , j ) = ( ( a + b ) / ( 1 + t f i ( 0 ) f j ( 0 ) ) , i + j ) , where f ( x ) = ( s x + 1 ) / ( t x + 1 ) , for appropriate parameters s , t ∈ ℤ m * . Each such loop is coupled to a 2-cocycle (in the group-theoretical sense) and this connection makes it possible to prove that the loop possesses a metacyclic inner mapping group. If s = 1 , then the loop is an A-loop. Questions of isotopism and isomorphism are considered in detail.

A computer algebra solution to a problem in finite groups.

Gert-Martin Greuel (2003)

Revista Matemática Iberoamericana

We report on a partial solution of the conjecture that the class of finite solvable groups can be characterised by 2-variable identities. The proof requires pieces from number theory, algebraic geometry, singularity theory and computer algebra. The computations were carried out using the computer algebra system SINGULAR.

A note on the minimal normal Fitting class

Marco Barlotti (1984)

Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni

Un gruppo finito ciclico-per-nilpotente appartiene alla minima classe di Fitting normale se e solo se è nilpotente.

A note on the Π -property of some subgroups of finite groups

Zhengtian Qiu, Guiyun Chen, Jianjun Liu (2024)

Czechoslovak Mathematical Journal

Let H be a subgroup of a finite group G . We say that H satisfies the Π -property in G if for any chief factor L / K of G , | G / K : N G / K ( H K / K ∩ L / K ) | is a π ( H K / K ∩ L / K ) -number. We obtain some criteria for the p -supersolubility or p -nilpotency of a finite group and extend some known results by concerning some subgroups that satisfy the Π -property.

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