Finite $2$-groups of class $2$ with a specific automorphism group

Document Type : Research Paper


1 University of Isfahan

2 Shahid Chamran university of Ahvaz


‎‎In this paper we determine all finite $2$-groups of‎ ‎class $2$ in which every automorphism of order $2$ leaving the Frattini subgroup elementwise fixed is inner‎.


Main Subjects

[1] A. Ab dollahi, Powerful p-groups have non-inner automorphisms of order p and some cohomology, J. Algebra, 323 (2010) 779-789.
[2] A. Ab dollahi, Finite p-groups of class 2 have noninner automorphisms of order p , J. Algebra, 312 (2007) 876-879.
[3] A. Ab dollahi and S. M. Ghoraishi, Noninner automorphisms of nite p -groups leaving the center elementwise xed, Int. J. Group Theory, 2 (2013) 17-20.
[4] A. Ab dollahi and S. M. Ghoraishi, On noninner 2-automorphisms of nite 2-groups, Bul l. Aust. Math. Soc., 90 (2014) 227-231.
[5] A. Ab dollahi, S. M. Ghoraishi, Y. Guerb oussa, M. Reguiat and B. Wilkens, Noninner automorphisms of order p for nite p-groups of co class 2, J. Group Theory, 17 (2014) 267-272.
[6] Y. Cheng, On nite p -groups with cyclic commutator subgroup, Arch. Math. (Basel), 39 (1982) 295-298.
[7] W. Gasch utz, Nichtab elsche p-Grupp en b esitzen aussere p -automorphismen, J. Algebra, 4 (1966) 1-2.
[8] D. Gorenstein, Finite Groups, Harp er & Row, New York, 1968.
[9] S. M. Ghoraishi, On noninner automorphisms of nite nonab elian p -groups, Bul l. Aust. Math. Soc., 89 (2014) 202{209.
[10] Y. K. Leong, Finite 2-groups of class two with cyclic centre, J. Austral. Math. Soc. (Ser. A), 27 (1979) 125{140.
[11] H. Lieb eck, Outer automorphisms in nilp otent p -groups of class 2, J. London Math. Soc., 40 (1965) 268-275.