A Proof of the Frattini Argument about Normal Series

Alexander O. Ilo, Oranugo Obinna Dominic and Egwu Andrew Onyeka
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Alexander O. Ilo: Department of Mathematics, Nnamdi Azikiwe University, P.M.B. 5025, Awka, Anambra State
Oranugo Obinna Dominic: Department of Mathematics, Nnamdi Azikiwe University, P.M.B. 5025, Awka, Anambra State
Egwu Andrew Onyeka: University of Nigeria, Nsukka

International Journal of Research and Scientific Innovation, 2025, vol. 12, issue 7, 2181-2187

Abstract: The Frattini Argument has notoriously many different stories and versions that a casual researcher may get confused about what this important statement is really all about. We are not going to discuss the different versions here; we want to supply a proof of one version that does not yet have a proof today. Every version of the Frattini Argument is attributed to the Italian mathematician Giovanni Frattini (1852-1925). The specific version of Frattini Argument that is our focus is concerned with the existence of a Series of Normal Subgroups below a finite group. Important as this theorem is, it has however not been proved by anyone, including Frattini himself. Although it has been generally accepted to be true—and used in analysis—no one has actually supplied a comprehensive proof of this theorem.

Date: 2025
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