Hostname: page-component-7479d7b7d-m9pkr Total loading time: 0 Render date: 2024-07-11T05:26:02.135Z Has data issue: false hasContentIssue false
  • English
  • Français

ON p-AUTOMORPHISMS THAT ARE INNER

Part of: Representation theory of groups

Published online by Cambridge University Press:  08 June 2009

M. SHABANI ATTAR
M. SHABANI ATTAR*
Affiliation:
Faculty of Science, Department of Mathematics, University of Payam Noor, Mashad, 91735-433, Iran (email: mehdishabani9@yahoo.com, m_shabaniattar@pnu.ac.ir)
*
✉For correspondence; e-mail: 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 group and let CAutΦ(G)(Z(Φ(G))) be the set of all automorphisms of G centralizing G/Φ(G) and Z(Φ(G)). For each prime p and finite p-group G, we prove that CAutΦ(G)(Z(Φ(G)))≤Inn(G) if and only if G is elementary abelian or Φ(G)=Z(G) and Z(G) is cyclic.

Keywords

automorphismcentral automorphismfinite p-group

MSC classification

Secondary: 20D45: Automorphisms
Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 80 , Issue 2 , October 2009 , pp. 291 - 293
Copyright
Copyright © Australian Mathematical Publishing Association Inc. 2009

References

[1] Curran, M. J., ‘Finite groups with central automorphism group of minimal order’, Math. Proc. R. Ir. Acad. 104A(2) (2004), 223–229.Google Scholar
[2] Curran, M. J. and McCaughan, D. J., ‘Central automorphisms that are almost inner’, Comm. Algebra 29(5) (2001), 2081–2087.Google Scholar
[3] Müller, O., ‘On p-automorphisms of finite p-groups’, Arch. Math. (Basel) 32 (1979), 533–538.CrossRefGoogle Scholar
[4] Robinson, D. J. S., A Course in the Theory of Groups (Springer, New York, 1982).CrossRefGoogle Scholar
[5] Schmid, P., ‘Frattinian p-groups’, Geom. Dedicata 36 (1990), 359–364.CrossRefGoogle Scholar
[6] Shabani Attar, M., ‘On central automorphisms that fix the centre elementwise’, Arch. Math. (Basel) 89 (2007), 296–297.CrossRefGoogle Scholar