Abdollahi, A., Ghoraishi, S. (2013). Noninner automorphisms of finite $p$-groups leaving the center elementwise fixed. International Journal of Group Theory, 2(4), 17-20. doi: 10.22108/ijgt.2013.2761

Alireza Abdollahi; S. Mohsen Ghoraishi. "Noninner automorphisms of finite $p$-groups leaving the center elementwise fixed". International Journal of Group Theory, 2, 4, 2013, 17-20. doi: 10.22108/ijgt.2013.2761

Abdollahi, A., Ghoraishi, S. (2013). 'Noninner automorphisms of finite $p$-groups leaving the center elementwise fixed', International Journal of Group Theory, 2(4), pp. 17-20. doi: 10.22108/ijgt.2013.2761

Abdollahi, A., Ghoraishi, S. Noninner automorphisms of finite $p$-groups leaving the center elementwise fixed. International Journal of Group Theory, 2013; 2(4): 17-20. doi: 10.22108/ijgt.2013.2761

Noninner automorphisms of finite $p$-groups leaving the center elementwise fixed

A longstanding conjecture asserts that every finite nonabelian $p$-group admits a noninner automorphism of order $p$. Let $G$ be a finite nonabelian $p$-group. It is known that if $G$ is regular or of nilpotency class $2$ or the commutator subgroup of $G$ is cyclic, or $G/Z(G)$ is powerful, then $G$ has a noninner automorphism of order $p$ leaving either the center $Z(G)$ or the Frattini subgroup $\Phi(G)$ of $G$ elementwise fixed. In this note, we prove that the latter noninner automorphism can be chosen so that it leaves $Z(G)$ elementwise fixed.

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