Hostname: page-component-7dd5485656-glrdx Total loading time: 0 Render date: 2025-10-30T19:40:54.610Z Has data issue: false hasContentIssue false
  • English
  • Français

EXISTENCE OF NONINNER AUTOMORPHISMS OF ORDER p IN SOME FINITE p-GROUPS

Part of: Representation theory of groups

Published online by Cambridge University Press:  17 September 2012

M. SHABANI-ATTAR
M. SHABANI-ATTAR*
Affiliation:
Department of Mathematics, Payame Noor University, P.O. Box 19395-3697 Tehran, Iran (email: mehdishabani9@yahoo.com, m_shabaniattar@pnu.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 G be a nonabelian finite p-group of order pm. A long-standing conjecture asserts that G admits a noninner automorphism of order p. In this paper we prove the validity of the conjecture if exp (G)=pm−2. We also show that if G is a finite p-group of maximal class, then G has at least p(p−1)noninner automorphisms of order p which fix Φ(G) elementwise.

Keywords

noninner automorphisms of order pp-groups

MSC classification

Secondary: 20D45: Automorphisms

Information

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 87 , Issue 2 , April 2013 , pp. 272 - 277
Copyright
©2012 Australian Mathematical Publishing Association Inc.

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]Abdollahi, A., Ghoraishi, M. & Wilkens, B., ‘Finite p-groups of class 3 have noninner automorphisms of order p’, Contrib. Algebra Geometry, to appear.Google Scholar
[4]Blackburn, N., ‘On a special class of p-groups’, Acta. Math. 100 (1958), 45–92.CrossRefGoogle Scholar
[5]Deaconescu, M. & Silberberg, G., ‘Noninner automorphisms of order p of finite p-groups’, J. Algebra 250 (2002), 283–287.CrossRefGoogle Scholar
[6]Huppert, B., Endliche Gruppen I (Springer, Berlin, 1967).CrossRefGoogle Scholar
[7]Jamali, A. R. & Viseh, M., ‘On the existence of noninner automorphisms of order two in finite 2-groups’, Bull. Aust. Math. Soc. 87 (2013), 278–287.CrossRefGoogle Scholar
[8]Liebeck, H., ‘Outer automorphisms in nilpotent p-groups of class 2’, J. Lond. Math. Soc. 40 (1965), 268–275.CrossRefGoogle Scholar
[9]Mazurov, V. D. & Khukhro (Eds.), E. I., ‘Unsolved problems in group theory’, in: The Kourovka Notebook, Vol. 16 (Russian Academy of Sciences, Siberian Division, Institue of Mathematics, Novosibirisk, 2006).Google Scholar
[10]Ninomiya, Y., ‘Finite p-groups with cyclic subgoups of index p 2’, Math. J. Okayama Univ. 36 (1994), 1–21.Google Scholar
[11]Shabani-Attar, M., ‘On a conjecture about automorphisms of finite p-groups’, Arch. Math. 93 (2009), 399–403.CrossRefGoogle Scholar
[12]Shabani-Attar, M., ‘A necessary condition for nonabelian finite p-groups with second centre of order p 2’, Indian J. Pure Appl. Math. 42(3) (2011), 183–186.CrossRefGoogle Scholar