Let $C_{Aut_{c}(G)}(Z(G))$ be the group of all central automorphisms of finite non abelian $p$-group $G$ fixing $Z(G)$ element wise. If $C_{Aut_{c}(G)}(Z(G))$ is a proper subgroup of $Inn(G)$, then what can we say about $G$? I know that $G$ is not of class two.
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1$\begingroup$ It would be good to remind people that an automorphism $\phi$ of $G$ is a central automorphism if $\phi(x)x^{-1}\in Z(G)$ for all $x\in G$. $\endgroup$– Arturo MagidinCommented Jul 4, 2017 at 13:16
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$\begingroup$ If $C_{{\rm Aut}_c(G)}(Z(G))={\rm Aut}_c(G)$, as is the case when $Z(G)$ has order 2, then there is nothing more that you can say. To see this choose $\phi\in{\rm Inn}(G)\setminus{\rm Aut}_c(G)$. If $\phi$ is conjugation by $g$, then there exists an $x\in G$ such that $x^gx^{-1}\not\in Z(G)$, because $\phi$ is not a central automorphism. This says that $G$ does not have nilpotency class 2. Hence you should assume $C_{{\rm Aut}_c(G)}(Z(G))$ is a proper subgroup of both ${\rm Aut}_c(G)$ and ${\rm Inn}(G)$. $\endgroup$– GlasbyCommented Jul 9, 2017 at 18:55
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$\begingroup$ It is not difficult to see that $Z(Inn(G)) \leq C_{Aut_{c}(G)}(Z(G)) \leq Aut_{c}(G)$ (see "FINITE GROUPS WITH CENTRAL AUTOMORPHISM GROUP OF MINIMAL ORDER"). Also we can check that if C_{Aut_{c}(G)}(Z(G)) wants to be a subgroup of $Inn(G)$ it should be equal to Z(Inn(G)) ... Thus I can ask my question in this way "If Z(Inn(G)) = C_{Aut_{c}(G)}(Z(G)) then what can I say about $G$?" $\endgroup$– banooCommented Jul 9, 2017 at 20:15
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