## Taiwanese Journal of Mathematics

### A COMPARISON OF THE ORDER COMPONENTS IN FROBENIUS AND 2-FROBENIUS GROUPS WITH FINITE SIMPLE GROUPS

#### Abstract

Let $G$ be a finite group. Based on the Gruenberg-Kegel graph ${\rm GK}(G)$, the order of $G$ can be divided into a product of coprime positive integers. These integers are called the order components of $G$ and the set of order components is denoted by ${\rm OC}(G)$. In this article we prove that, if $S$ is a non-Abelian finite simple group with a disconnected graph ${\rm GK}(S)$, with an exception of $U_4(2)$ and $U_5(2)$, and $G$ is a finite group with ${\rm OC}(G)={\rm OC}(S)$, then $G$ is neither Frobenius nor $2$-Frobenius. For a group $S$ isomorphic to $U_4(2)$ or $U_5(2)$, we construct examples of $2$-Frobenius groups $G$ such that ${\rm OC}(S)={\rm OC}(G)$. In particular, the simple groups $U_4(2)$ and $U_5(2)$ are not recognizable by their order components.

#### Article information

Source
Taiwanese J. Math., Volume 13, Number 1 (2009), 67-89.

Dates
First available in Project Euclid: 18 July 2017

https://projecteuclid.org/euclid.twjm/1500405273

Digital Object Identifier
doi:10.11650/twjm/1500405273

Mathematical Reviews number (MathSciNet)
MR2489308

Zentralblatt MATH identifier
1230.20013

Subjects
Primary: 20D05: Finite simple groups and their classification

#### Citation

Moghaddamfar, A. R. A COMPARISON OF THE ORDER COMPONENTS IN FROBENIUS AND 2-FROBENIUS GROUPS WITH FINITE SIMPLE GROUPS. Taiwanese J. Math. 13 (2009), no. 1, 67--89. doi:10.11650/twjm/1500405273. https://projecteuclid.org/euclid.twjm/1500405273

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