On the Probabilistic Zeta Function for Finite Groups

John Shareshian

doi:10.1006/jabr.1998.7560

On the Probabilistic Zeta Function for Finite Groups

John Shareshian

Research output: Contribution to journal › Article › peer-review

5 Scopus citations

Abstract

LetGbe a finite group, and define the function[formula]where μ is the Möbius function on the subgroup lattice ofG. The functionP(G,s) is the multiplicative inverse of a zeta function forG, as described by Mann and Boston. Boston conjectured thatP′(G,1)=0 ifGis a nonabelian simple. We will prove a generalization of this conjecture, showing thatP′(G,1)=0 unlessG/O_p(G) is cyclic for some primep.

Original language	English
Pages (from-to)	703-707
Number of pages	5
Journal	Journal of Algebra
Volume	210
Issue number	2
DOIs	https://doi.org/10.1006/jabr.1998.7560
State	Published - Dec 15 1998

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10.1006/jabr.1998.7560

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abstract = "LetGbe a finite group, and define the function[formula]where μ is the M{\"o}bius function on the subgroup lattice ofG. The functionP(G,s) is the multiplicative inverse of a zeta function forG, as described by Mann and Boston. Boston conjectured thatP′(G,1)=0 ifGis a nonabelian simple. We will prove a generalization of this conjecture, showing thatP′(G,1)=0 unlessG/Op(G) is cyclic for some primep.",

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AB - LetGbe a finite group, and define the function[formula]where μ is the Möbius function on the subgroup lattice ofG. The functionP(G,s) is the multiplicative inverse of a zeta function forG, as described by Mann and Boston. Boston conjectured thatP′(G,1)=0 ifGis a nonabelian simple. We will prove a generalization of this conjecture, showing thatP′(G,1)=0 unlessG/Op(G) is cyclic for some primep.

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On the Probabilistic Zeta Function for Finite Groups

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