Research Projects

Interest in the equality of two different automorphism groups dates back to 1908, when Hilton \cite[p. 233]{hilton} asked the following question: Whether a non-abelian group can have an abelian group of isomorphisms (automorphisms), that is, when $\Aut(G)=\Cent(G)$? Therefore, it becomes interesting topic to study when two different automorphism groups of a group are equal or isomorphic. For more details about this problem, one can see the survey article by Yadav \cite{yadavsurvey}. Also there is a longstanding conjecture which asserts that every finite non-abelian $p$-group admits a non-inner automorphism of order $p$ (see also \cite[Problem $4.13$]{maz}). Infact, some researchers are intrested in proving the sharpened version of the conjucture. They are intreseted in proving that every finite non-abelian $p$-group $G$ has a non-inner automorphism of order $p$ which fixes $\Phi(G)$ elementwise.