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‎‎‎‎‎‎‎Let ‎$‎G‎$ ‎be a‎ ‎group ‎and ‎‎$‎‎\alpha\in Aut(G)$ ‎be a‎ ‎fixed ‎automorphism ‎of ‎‎$‎G‎$‎. An ‎$‎‎\alpha$-commutator ‎of ‎elements ‎‎$‎‎x, y\in G$ ‎is ‎‎$‎‎[x, y]_{\alpha}= x^{-1}y^{-1}xy^{\alpha}$. ‎The ‎‎$‎‎\alpha$-center ‎subgroup ‎of ‎‎$‎G‎$‎, denoted by ‎$‎‎Z^{\alpha}(G)$ ‎is ‎defined ‎as ‎‎$‎‎Z^{\alpha}(G)= \{x\in G : [y, x]_{\alpha}= 1, ‎\forall ‎y\in ‎G‎ \}‎‎$. ‎ ‎If ‎$‎‎N$ ‎is a‎ ‎normal ‎subgroup ‎of ‎‎$‎G‎$ ‎which ‎is ‎invariant ‎under ‎‎$‎‎\alpha$ ‎and‎ ‎‎$\bar{\alpha}‎$ is an automorphism of quotient group‎‎‎‎‎ ‎‎‎‎‎$‎G/N$ ‎such that ‎send ‎an ‎element ‎‎$‎‎gN$ ‎to ‎‎$‎‎g^{\alpha}N$‎, then ‎the ‎following normal ‎series‎ $$ ‎\{ ‎1\}= ‎G_{0}‎\unlhd ‎G_{1}‎\unlhd ‎\dots‎ ‎\unlhd ‎G_{n}= ‎G‎, ‎‎$$ ‎‎is called a central ‎$‎‎\alpha$‎-series whenever ‎$‎‎G_{i}^{\alpha}= G_{i}$ ‎and ‎‎$‎‎G_{i+1}/G_{i}‎\leq Z^{\bar{\alpha}}(G/G_{i})‎$‎, for ‎$‎‎0‎\leq i‎\leq n-1‎‎$‎.‎‎ ‎An ‎‎$‎‎\alpha$-nilpotent ‎group ‎is a‎ ‎group ‎which ‎possesses ‎at ‎least a‎ ‎central ‎‎$‎‎‎\alpha‎$‎-series. It is to see that f $\alpha$ is an inner automorphism of a nilpotent group $G$, then $G$ is an $\alpha$-nilpotent group. My Question is: Is there any non-inner automorphim $\alpha$ of a finite non-abelian $p$-group $P$, such that $P$ is $\alpha$-nilpotent?

Arturo Magidin
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Banoo
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  • this file may be useful for answering this question http://cjms.journals.umz.ac.ir/article_824_4759576cb48aca7d4f224a1d86125532.pdf – Banoo Sep 04 '20 at 20:27
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    $Z^{\alpha}(G)$ is the intersection of $Z(G)$ and the fixed set of $\alpha$. – Arturo Magidin Sep 04 '20 at 22:43
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    What about $G=C_2\times C_2$, $C_2$ cyclic of order $2$, and $\alpha(x,y)=(y,x)$? – YCor Sep 05 '20 at 09:49
  • Thanks for your comment. Actually I've answered my question for all finitely generated abelian groups (non-inner= non-identity). Now my problem is about non-abelian groups. I've answered my question for some classes of finite non-abelian p-groups but i can not answer it in general. I think the answer is "yes" for all non-abelian groups. @YCor – Banoo Sep 05 '20 at 13:26
  • Can you quantify your question? "of a finite non-abelian" is unclear to me, do you mean "of some finite non-abelian", "of every some finite non-abelian", or anything else? – YCor Sep 05 '20 at 13:40
  • Let $P$ be an arbitrary finite $p$-group. Is there any "non-inner" automorphism in $Aut(P)$, such that $P$ is an $\alpha$-nilpotent group? I could find such non-inner autmorphism for $p$-groups of order $p^3$ or $p^4$, or $p^5$ and nilpotent p-groups of nilpotency class 2 with non-cyclic center... – Banoo Sep 05 '20 at 16:41

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