Automorphisms of prime order almost regular in the sense of rank

Abstract

Several finite groups admitting automorphisms of prime order which are almost regular in the sense of rank are presented. Three theorems are presented in this context, in which the first theorem describes that if a finite nilpotent group G admits an automorphism I¦ of prime order p with fixed-point sub group CG(I¦) of rank r, then G has a characteristic subgroup C such that its nilpotency class is p-bounded and the quotient group G/C has (p,r)-bounded rank. The second theorem presents that a group G contains a nilpotent periodic normal subgroup H of nilpotency class c for which the quotient group G/H has finite rank r. While, the third theorem provides that if a finite nilpotent group G of derived length d admits an automorphism of prime order p with centralizer of rank r, then the group G has a characteristic subgroup C such that its nilpotency class is p-bounded and the quotient group G/C has (p,r,d)-bounded rank.