Hostname: page-component-6bf8c574d5-vmclg Total loading time: 0 Render date: 2025-02-19T04:58:56.262Z Has data issue: false hasContentIssue false
  • English
  • Français

A NOTE ON AUTOMORPHISMS OF FINITE p-GROUPS

Part of: Representation theory of groups

Published online by Cambridge University Press:  30 March 2012

S. MOHSEN GHORAISHI
S. MOHSEN GHORAISHI*
Affiliation:
Department of Mathematics, University of Isfahan, Isfahan 81746-73441, Iran (email: m.ghoraishi@sci.ui.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 p be an odd prime and let G be a finite p-group such that xZ(G)⊆xG, for all x∈G∖Z(G), where xG denotes the conjugacy class of x in G. Then G has a noninner automorphism of order p leaving the Frattini subgroup Φ(G) elementwise fixed.

Keywords

finite p-groupsautomorphisms of p-groupsnoninner automorphismsCamina pairs

MSC classification

Secondary: 20D15: Nilpotent groups, $p$-groups
Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 87 , Issue 1 , February 2013 , pp. 24 - 26
Copyright
Copyright © Australian Mathematical Publishing Association Inc. 2012

References

[1]Abdollahi, A., ‘Finite p-groups of class 2 have noninner automorphisms of order p’, J. Algebra 312 (2007), 876–879.CrossRefGoogle Scholar
[2]Abdollahi, A., ‘Powerful p-groups have noninner automorphisms of order p and some cohomology’, J. Algebra 323 (2010), 779–789.CrossRefGoogle Scholar
[3]Deaconescu, M. and Silberberg, G., ‘Noninner automorphisms of order p of finite p-groups’, J. Algebra 250 (2002), 283–287.CrossRefGoogle Scholar
[4]Gaschütz, W., ‘Nichtabelsche p-Gruppen besitzen äussere p-Automorphismen’, J. Algebra 4 (1966), 1–2.CrossRefGoogle Scholar
[5]Gorenstein, D., Finite Groups (Harper & Row, New York, 1968).Google Scholar
[6]Liebeck, H., ‘Outer automorphisms in nilpotent p-groups of class 2’, J. Lond. Math. Soc. (2) 40 (1965), 268–275.CrossRefGoogle Scholar
[7]Macdonald, I. D., ‘Some p-groups of Frobenius and extra-special type’, Israel J. Math. 40 (1981), 350–364.CrossRefGoogle Scholar
[8]Mazurov, V. D. and Khukhro (ed.), E. I., ‘Unsolved problems in group theory’, in: The Kourovka Notebook, No. 16 (Russian Academy of Sciences, Siberian Division, Institue of Mathematics, Novosibirisk, 2006).Google Scholar
[9]Schmid, P., ‘A cohomological property of regular p-groups’, Math. Z. 175 (1980), 1–3.CrossRefGoogle Scholar
[10]Yadav, M. K., ‘On automorphisms of finite p-groups’, J. Group Theory 10 (2007), 859–866.CrossRefGoogle Scholar