On equality of coset preserving subcentral automorphisms

Let G be a finite non-abelian p -group, where p is a prime and M and N are two subcentral characteristic subgroups of G . An automorphism α of G is called subcentral automorphism if, for all g ∈ G , g − 1 α ( g ) ∈ M and for all n ∈ N , n − 1 α ( n ) = 1 . Let Aut MN ( G ) , C Aut MN ( G ) ( Z ( G )) and Aut G ′ N ( G ) denote, respectively, the group of all subcentral automorphisms of G , the group of all subcentral automorphisms of G fixing the center of G , elementwise, and the group of all derival automorphisms of G fixing the elements of N . In this study, we present necessary and sufficient conditions on a finite p -group, G , such that Aut MN ( G ) = C Aut MN ( G ) ( Z ( G )) and Aut MN ( G ) = Aut G ′ N ( G ) . Moreover, we investigate the necessary and sufficient conditions for the equality of inner automorphisms and the group of subcentral automorphisms that fix the center and the Frattini subgroup of, G , element wise.