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On equality of central and class preserving automorphisms of finite \(p\)-groups. (English) Zbl 1297.20021

An automorphism \(\alpha\) of a group \(G\) is called a class preserving automorphism if \(\alpha(x)\in x^G\) for all \(x\in G\), where \(x^G\) is the conjugacy class of \(x\); and an automorphism \(\varphi\) is called a central automorphism if \(g^{-1}\varphi(g)\in Z(G)\) for all \(g\in G\). Let \(\operatorname{Aut}_c(G)\) denote the set of all class preserving automorphisms and let \(\operatorname{Aut}_z(G)\) denote the set of all central automorphisms.
The authors prove that in a finite non-Abelian \(p\)-group, \(\operatorname{Aut}_c(G)=\operatorname{Aut}_z(G)\) if and only if \(\operatorname{Aut}_c(G)=\operatorname{Hom}(G/Z(G),\gamma_2(G))\) and \(\gamma_2(G)=Z(G)\), where \(Z(G)\) is the center of \(G\) and \(\gamma_2(G)=[G,G]\) the derived subgroup of \(G\). They also classify all finite \(p\)-groups \(G\) of class 2 such that \(\operatorname{Aut}_c(G)=\operatorname{Aut}_z(G)\) and characterize all finite \(p\)-groups of order \(\leq p^7\) with the condition \(\operatorname{Aut}_c(G)=\operatorname{Aut}_z(G)\).

MSC:

20D45 Automorphisms of abstract finite groups
20D15 Finite nilpotent groups, \(p\)-groups
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References:

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