The multiplicator of finite nilpotent groups

Abstract

Let G be a group and M a G -module; then d ( G ) denotes the minimal number of generators of G and d G ( M ) the minimal number of generators over ZG of M . For G a finite nilpotent group let G = F / R , F free, be a presentation for G ; then it is shown that d ( R /[ F , R ]) = d G (R/[ R , R ]), that is d ( G ) + d ( M ( G )) = d G ( R / R ′), where M ( G ) denotes the Schur multiplicator of G .