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SOME PROPERTIES OF SEMI-ABELIAN $p$-GROUPS

Part of: Representation theory of groups Connections with homological algebra and category theory

Published online by Cambridge University Press:  15 October 2014

MOHAMMED T. BENMOUSSA  and
YASSINE GUERBOUSSA
MOHAMMED T. BENMOUSSA
Affiliation:
Department of Mathematics, University Kasdi Merbah Ouargla, Ouargla, Algeria email [email protected]
YASSINE GUERBOUSSA*
Affiliation:
Department of Mathematics, University Kasdi Merbah Ouargla, Ouargla, Algeria email [email protected]
*
✉[email protected]
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Abstract

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We prove a cohomological property for a class of finite $p$-groups introduced earlier by Xu, which we call semi-abelian $p$-groups. This result implies that a semi-abelian $p$-group has noninner automorphisms of order $p$, which settles a long-standing problem for this class. We answer also, independetly, an old question posed by Xu about the power structure of semi-abelian $p$-groups.

Keywords

finite p-groupscohomology of groupsautomorphisms

MSC classification

Primary: 20D15: Nilpotent groups, $p$-groups
Secondary: 20J06: Cohomology of groups
Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 91 , Issue 1 , February 2015 , pp. 86 - 91
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.Google Scholar
Abdollahi, A., ‘Cohomologically trivial modules over finite groups of prime power order’, J. Algebra 342 (2011), 154–160.Google Scholar
Berkovich, Y., Groups of Prime Power Order, Vol. 1 (Walter de Gruyter, Berlin, 2008).Google Scholar
Bubboloni, D. and Corsi Tani, G., ‘p-groups with all the elements of order p in the center’, Algebra Colloq. 11 (2004), 181–190.Google Scholar
Deaconescu, M. and Silberberg, G., ‘Noninner automorphisms of order p of finite p-groups’, J. Algebra 250 (2002), 283–287.Google Scholar
Dixon, J., du Sautoy, M., Mann, A. and Segal, D., Analytic Pro-p Groups, 2nd edn (Cambridge University Press, Cambridge, 1999).Google Scholar
Ghoraishi, M. S., ‘On noninner automorphisms of finite nonabelian p-groups’, Bull. Aust. Math. Soc. 89 (2014), 202–209.CrossRefGoogle Scholar
Gruenberg, K. W., Cohomological Topics in Group Theory, Lecture Notes in Mathematics, 143 (Springer, Berlin, 1970).Google Scholar
Hoechsmann, K., Roquette, P. and Zassenhaus, H., ‘A cohomological characterization of finite nilpotent groups’, Arch. Math. 19 (1968), 225–244.CrossRefGoogle Scholar
Huppert, B., Endliche gruppen I, Grundlehren der Mathematischen Wissenschaften, Band 134 (Springer, Berlin, 1967).CrossRefGoogle Scholar
Mazurov, V. D. and Khukhro, E. I., The Kourovka Notebook. Unsolved Problems in Group Theory, 18th edn (Russian Academy of Sciences, Siberian Division, Institute of Mathematics, Novosibirsk, 2014).Google Scholar
Schmid, P., ‘A cohomological property of regular p-groups’, Math. Z. 175 (1980), 1–3.Google Scholar
Xu, M. Y., ‘A class of semi-p-abelian p-groups’, Kexue Tongbao 26 (1981), 453–456 (in Chinese); translation in Kexue Tongbao 27 (1982), 142–146 (English Ed.).Google Scholar
Xu, M. Y., ‘The power structure of finite p-groups’, Bull. Aust. Math. Soc. 36(1) (1987), 1–10.Google Scholar