Fatemeh Vosooghpour, Zeinab Kargarian, and Mehri Akhavan-Malayeri Alzahra University, Alzahra University, Alzahra University

Abstract : Let $G$ be a group and let $p$ be a prime number. If the set $\A(G)$ of all commuting automorphisms of $G$ forms a subgroup of $\Aut(G)$, then $G$ is called $\A(G)$-group. In this paper we show that any $p$-group with cyclic maximal subgroup is an $\A(G)$-group. We also find the structure of the group $\A(G)$ and we show that $\A(G)=\Aut_c(G)$. Moreover, we prove that for any prime $p$ and all integers $n\geq3$, there exists a non-abelian $\A(G)$-group of order $p^n$ in which $\A(G)=\Aut_c(G)$. If $p>2$, then $\A(G)\cong\Z_p\times\Z_{p^{n-2}}$ and if $p=2$, then $\A(G)\cong\Z_2\times\Z_2\times\Z_{2^{n-3}}$ or $\Z_2\times\Z_2$.