The maximal normal 𝑝-subgroup of the automorphism group of an abelian 𝑝-group

Abstract

Let p p be a prime number and let G \,G\, be an abelian p p –group. Let Δ \Delta be the maximal normal p p –subgroup of Aut ⁡ G \operatorname {Aut}G and ζ \zeta the maximal p p –subgroup of its centre. Let t \mathbf {t} be the torsion radical of E ( G ) {\mathcal {E}}(G) . Then Δ = ( 1 + t ) ζ \Delta =(1+\mathbf {t})\zeta . The result is new for p = 2 p=2 and 3, and the proof is new and valid for all primes p p .