Hostname: page-component-cd9895bd7-mkpzs Total loading time: 0 Render date: 2024-12-26T04:40:08.292Z Has data issue: false hasContentIssue false
  • English
  • Français

ON NONINNER 2-AUTOMORPHISMS OF FINITE 2-GROUPS

Part of: Representation theory of groups Structure and classification of infinite or finite groups

Published online by Cambridge University Press:  29 May 2014

ALIREZA ABDOLLAHI  and
S. MOHSEN GHORAISHI
ALIREZA ABDOLLAHI
Affiliation:
Department of Mathematics, University of Isfahan, Isfahan 81746-73441, Iran School of Mathematics, Institute for Research in Fundamental Sciences (IPM), PO Box 19395-5746, Tehran, Iran email a.abdollahi@math.ui.ac.ir
S. MOHSEN GHORAISHI*
Affiliation:
Department of Mathematics, Faculty of Mathematics and Computer Sciences, Shahid Chamran University, Ahvaz, Iran email m.ghoraishi@scu.ac.ir
*
✉m.ghoraishi@scu.ac.ir
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

Let $\def \xmlpi #1{}\def \mathsfbi #1{\boldsymbol {\mathsf {#1}}}\let \le =\leqslant \let \leq =\leqslant \let \ge =\geqslant \let \geq =\geqslant \def \Pr {\mathit {Pr}}\def \Fr {\mathit {Fr}}\def \Rey {\mathit {Re}}G$ be a finite 2-group. If $G$ is of coclass 2 or $(G,Z(G))$ is a Camina pair, then $G$ admits a noninner automorphism of order 2 or 4 leaving the Frattini subgroup elementwise fixed.

Keywords

finite p-groupnoninner automorphismcoclassCamina pair

MSC classification

Secondary: 20D45: Automorphisms 20E36: Automorphisms of infinite groups
Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 90 , Issue 2 , October 2014 , pp. 227 - 231
Copyright
Copyright © 2014 Australian Mathematical Publishing Association Inc. 

References

Abdollahi, A., ‘Powerful p-groups have noninner automorphisms of order p and some cohomology’, J. Algebra 323 (2010), 779–789.CrossRefGoogle Scholar
Abdollahi, A., ‘Finite p-groups of class 2 have noninner automorphisms of order p’, J. Algebra 312 (2007), 876–879.Google Scholar
Abdollahi, A. and Ghoraishi, S. M., ‘Noninner automorphisms of finite p-groups leaving the center elementwise fixed’, Int. J. Group Theory 2 (2013), 17–20.Google Scholar
Abdollahi, A., Ghoraishi, S. M., Guerboussa, Y., Reguiat, M. and Wilkens, B., ‘Noninner automorphisms of order p for finite p-groups of coclass 2’, J. Group Theory 17 (2014), 267–272.CrossRefGoogle Scholar
Abdollahi, A., Ghoraishi, M. and Wilkens, B., ‘Finite p-groups of class 3 have noninner automorphisms of order p’, Beitr. Algebra Geom. 54 (2013), 363–381.Google Scholar
Deaconescu, M. and Silberberg, G., ‘Noninner automorphisms of order p of finite p-groups’, J. Algebra 250 (2002), 283–287.CrossRefGoogle Scholar
Dekimpe, K. and Eick, B., ‘Computational aspects of group extensions and their application in topology’, Exp. Math. 11 (2002), 183–200.Google Scholar
Gaschütz, W., ‘Nichtabelsche p-Gruppen besitzen äussere p-Automorphismen’, J. Algebra 4 (1966), 1–2.CrossRefGoogle Scholar
Ghoraishi, S. M., ‘A note on automorphisms of finite p-groups’, Bull. Aust. Math. Soc. 87 (2013), 24–26.Google Scholar
Ghoraishi, S. M., ‘On noninner automorphisms of finite nonabelian p-groups’, Bull. Aust. Math. Soc. 89 (2014), 202–209.CrossRefGoogle Scholar
Jamalli, A. R. and Viseh, M., ‘On the existence of noninner automorphisms of order two in finite 2-groups’, Bull. Aust. Math. Soc. 87 (2013), 278–287.Google Scholar
Liebeck, H., ‘Outer automorphisms in nilpotent p-groups of class 2’, J. Lond. Math. Soc. 40 (1965), 268–275.Google Scholar
Mazurov, V. D. and Khukhro, E. I. (eds.) ‘Unsolved problems in group theory’, in: The Kourovka Notebook, Vol. 16 (Russian Academy of Sciences, Siberian Division, Institue of Mathematics, Novosibirisk, 2006).Google Scholar
Schmid, P., ‘A cohomological property of regular p-groups’, Math. Z. 175 (1980), 1–3.Google Scholar
Shabani-Attar, M., ‘Existence of noninner automorphisms of order p in some finite p-groups’, Bull. Aust. Math. Soc. 87 (2013), 272–277.Google Scholar
Sims, C. C., Computation with Finitly Presented Groups (Cambridge University Press, Cambridge, 1994).Google Scholar