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

Document Type: Research Paper

Authors

University of Isfahan

Abstract

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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A. Abdollahi (2010). Powerful $p$-groups have noninner automorphisms of order $p$ and some cohomology. J. Algebra. 323, 779-789
A. Abdollahi (2007). Finite $p$-groups of class $2$ have noninner automorphisms of order $p$. J. Algebra. 312, 876-879
A. Abdollahi, M. Ghoraishi and B. Wilkens (2013). Finite $p$-groups of class 3 have noninner automorphisms of order $p$. Beitr. Algebra Geom.. 54 (1), 363-381
M. Deaconescu and G. Silberberg (2002). Noninner automorphisms of order $p$ of finite $p$-groups. J. Algebra. 250, 283-287
W. Gasch"{u}tz (1966). Nichtabelsche $p$-Gruppen besitzen "{a}ussere $p$-Automorphismen. J. Algebra. 4, 1-2
A. R. Jamali and M. Viseh (2013). On the existence of noinner automorphisms of order two in finite $2$-groups. Bull. Aust. Math. Soc.. 87 (2), 278-287
H. Liebeck (1965). Outer automorphisms in nilpotent $p$-groups of class $2$. J. London Math. Soc.. 40, 268-275
P. Schmid (1976). Normal $p$-subgroups in the group of outer automorphisms of a finite $p$-group. Math. Z.. 147 (3), 271-277
P. Schmid (1980). A cohomological property of regular $p$-groups. Math. Z.. 175, 1-3