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ON THE EXISTENCE OF NONINNER AUTOMORPHISMS OF ORDER TWO IN FINITE 2-GROUPS

Part of: Representation theory of groups

Published online by Cambridge University Press:  17 September 2012

A. R. JAMALI  and
M. VISEH
A. R. JAMALI*
Affiliation:
Faculty of Mathematical Sciences and Computer, Tarbiat Moallem University, 599 Taleghani Ave., Tehran 15618, Iran (email: jamali@tmu.ac.ir)
M. VISEH
Affiliation:
Faculty of Mathematical Sciences and Computer, Tarbiat Moallem University, 599 Taleghani Ave., Tehran 15618, Iran (email: m.viseh@tmu.ac.ir)
*
✉For correspondence; e-mail: jamali@tmu.ac.ir
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Abstract

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In this paper we prove that every nonabelian finite 2-group with a cyclic commutator subgroup has a noninner automorphism of order two fixing either Φ(G) or Z(G) elementwise. This, together with a result of Peter Schmid on regular p-groups, extends our result to the class of nonabelian finite p-groups with a cyclic commutator subgroup.

Keywords

finite p-groupsnoninner automorphismpowerful p-groupscyclic commutator subgroup

MSC classification

Secondary: 20D45: Automorphisms
Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 87 , Issue 2 , April 2013 , pp. 278 - 287
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]Berkovich, Y., Groups of Prime Power Order, Vol.1 (Walter de Gruyter, Berlin, 2008).Google Scholar
[4]Caranti, A. & Scoppola, C. M., ‘Endomorphisms of two-generated metabelian groups that induce the identity modulo the derived subgroup’, Arch. Math. 56 (1991), 218–227.CrossRefGoogle Scholar
[5]Cheng, Y., ‘On finite p-groups with cyclic commutator subgroup’, Arch. Math. 39 (1982), 295–298.CrossRefGoogle Scholar
[6]Deaconescu, M. & Silberberg, G., ‘Noninner automorphisms of order p of finite p-groups’, J. Algebra 250 (2002), 283–287.CrossRefGoogle Scholar
[7]Finogenov, A. A., ‘Finite p-groups with a cyclic commutator subgroup’, Algebra Logic 34(2) (1995), 125–129.CrossRefGoogle Scholar
[8]Gaschütz, W., ‘Nichtabelsche p-Gruppen besitzen äussere p-Automorphismen’, J. Algebra 4 (1966), 1–2.CrossRefGoogle Scholar
[9]Kurzweil, H. & Stellmacher, B., The Theory of Finite Groups: An Introduction (Springer, New York, 2004).CrossRefGoogle Scholar
[10]Liebeck, H., ‘Outer automorphisms in nilpotent p-groups of class 2’, J. Lond. Math. Soc. 40 (1965), 268–275.CrossRefGoogle Scholar
[11]Mazurov, V. D. & Khukhro (Eds.), E. I., ‘Unsolved problems in group theory’, in: The Kourovka Notebook, Vol. 16 (Russian Academy of Sciences, Siberian Division Institute of Mathematics, Novosibirsk, 2006).Google Scholar
[12]Schmid, P., ‘Frattinian p-groups’, Geom. Dedicata 36 (1990), 359–364.CrossRefGoogle Scholar
[13]Shabani-Attar, M., ‘Existence of noninner automorphisms of order p in some finite p-groups’, Bull. Aust. Math. Soc. 87 (2013), 272–277.CrossRefGoogle Scholar