Some properties of certain automorphism groups of finite \(p\)-groups (Q6650444)

Let \(G\) be a finite group and let \(L(G)=\{x \in G \mid x^{\alpha}=x \;\;\boldsymbol{\forall} \alpha \in \mathrm{Aut}(G) \}\) be the absolute center of \(G\). An automorphism \(\alpha \in \mathrm{Aut}(G)\) is called an absolute central automorphism if \([x,\alpha]=x^{-1}x^{\alpha} \in L(G)\) for every \(x \in G\). The author calls an automorphism \(\alpha \in \mathrm{Aut}(G)\) a \(n\)th autoclass-preserving automorphism if for all \(x\in G\), there exists an element \(g_{x}\in K_{n-1}(G)\) such that \(x\alpha= g_{x}^{-1}xg_{x}\), where \(K_{n-1}(G)\) is the \((n-1)\)th autocommutator subgroup of \(G\).\N\NIn this paper the author characterizes finite \(p\)-groups of class \(2\) such that every central automorphism is absolute central. He also obtains a necessary and sufficient condition for an autonilpotent finite \(p\)-group of class \(n+1\) such that each absolute central automorphism is a \(n\)th autoclass-preserving.