Document Type: Original Manuscript


Department of Mathematics, Payame Noor University (PNU), P.O.Box 19395-3697, Tehran, Iran.


Let $G$ be a finite non-abelian $p$-group and $L(G)$ denotes the absolute center of $G$. Also, let $\Aut^{L}(G)$ and $\Aut_c(G)$ denote the group of all absolute central and the class preserving automorphisms of $G$, respectively. In this paper, we give a necessary and sufficient condition for $G$ such that $\Aut_c(G)=\Aut^{L}(G)$. We also characterize all finite non-abelian $p$-groups of order $p^n (n\leq 5)$, for which every absolute central automorphism is class preserving.