Recognizing the Non-Frattini Abelian Chief Factors of a Finite Group from Its Probabilistic Zeta Function

Abstract

Given a finite group G, the Dirichlet polynomial of G is

The multiplicative counterpart of this polynomial is called the Probabilistic Zeta function of the group G. We prove that if H is a finite group such that P _G (s) = P _H (s), then G and H have the same non-Frattini abelian chief factors.

Key Words:

• Chief factors
• Dirichlet polynomial
• Probabilistic Zeta function

2010 Mathematics Subject Classification:

• Primary 20D30
• Secondary 11M41, 20D06, 20E28

Notes

Communicated by P. Tiep.