MATEMATIČKI VESNIK
МАТЕМАТИЧКИ ВЕСНИК



MATEMATIČKI VESNIK
Finite groups admitting some coprime operator groups
Enrico Jabara

Abstract

Let $G$ be a finite group, with a finite operator group $A$, satisfying the following conditions: (1)~$(\vert G \vert, \vert A \vert)=1$; (2)~there exists a natural number $m$ such that for any $ \alpha, \beta \in A^{\sharp}$ we have: $[\,C_G(\alpha),\underbrace{C_G(\beta),\dots,C_G(\beta)}_{m}\,]=\{1\}$; (3)~$A$ is not cyclic. We prove the following: (1)~If the exponent $n$ of $A$ is square-free, then $G$ is nilpotent and its class is bounded by a function depending only on $m$ and $\lambda(n)$ ($=n$). (2)~If $Z(A)=\{1\}$ and $A$ has exponent $n$, then $G$ is nilpotent and its class is bounded by a function depending only on $m$ and $\lambda(n)$.

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Keywords: Finite groups, operator group, Frobenius group.

MSC: 20D45

Pages:  31--37     

Volume  58 ,  Issue  1$-$2 ,  2006