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Paper   IPM / M / 9455
School of Mathematics
  Title:   A comparison of the order components in Frobenius and 2-Frobenius groups with finite simple groups
  Author(s):  A. R. Moghaddamfar
  Status:   Published
  Journal: Taiwanese J. Math.
  Vol.:  13
  Year:  2009
  Pages:   67-89
  Supported by:  IPM
  Abstract:
Let G be a finite group. Based on the Gruenberg-Kegel graph 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 OC(G). In this article we prove that, if S is a non-Abelian finite simple group with a disconnected graph GK(S), with an exception of U4(2) and U5(2), and G is a finite group with OC(G) = OC(S), then G is neither Frobenius nor 2-Frobenius. For a group S isomorphic to U4(2) or U5(2), we construct examples of 2-Frobenius groups G such that OC(S) = OC(G). In particular, the simple groups U4(2) and U5(2) are not recognizable by their order components.

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