Abstract
In this paper we study the existence of at least one non-inner automorphism of order p of a non-abelian finite p-group of coclass 3, for any prime \(p\ne 3\).
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References
Abdollahi, A.: Finite \(p\)-groups of class \(2\) have noninner automorphisms of order \(p\). J. Algebra 312, 876–879 (2007)
Abdollahi, A.: Powerful \(p\)-groups have noninner automorphism of order \(p\) and some cohomology. J. Algebra 323, 779–789 (2010)
Abdollahi, A., Ghoraishi, S.M., Wilkens, B.: Finite p-groups of class 3 have noninner automorphisms of order p. Beitr. Algebra Geom. 54, 363–381 (2013)
Abdollahi, A., Ghoraishi, S.M., Guerboussa, Y., Reguiat, M., Wilkens, B.: Noninner automorphism of order \(p\) for finite \(p\)-groups of coclass \(2\). J. Group Theory 17, 267–272 (2014)
Benmoussa, M.T., Guerboussa, Y.: Some properties of semi-abelian \(p\)-groups. Bull. Aust. Math. Soc. 91, 86–91 (2015)
Blackburn, N.: On a special class of \(p\)-groups. Acta Math. 100, 45–92 (1958)
Caranti, A., Scoppola, C.M.: Endomorphisms of two-generated metabelian groups that induce the identity modulo the derived subgroup. Arch. Math. 56, 218–227 (1991)
Deaconescu, M.: Noninner automorphisms of order \(p\) of finite \(p\)-groups. J. Algebra 250, 283–287 (2002)
Gaschutz, W.: Nichtabelsche \(p\)-Gruppen besitzen au\(\beta \)ere \(p\)-automorphismen. J. Algebra 4, 1–2 (1966)
Gavioli, N., Pannone, V.: Frobenius partitions in extraspecial \(p\)-groups. Geom. Dedic. 74, 313–323 (1999)
Gruenberg, K.W.: Cohomological methods in group theory, Lecture notes in Mathematics, p. 143. Springer Verlag, Berlin (1970)
Huppert, B.: Endliche Gruppen. Springer-Verlag, Berlin (1967)
Jamali, A.R., Viseh, R.: On the existence of noninner automorphism of order two in finite \(2\)-groups. Bull. Aust. Math. Soc. 87, 278–287 (2013)
Leedham-Green, C.R., McKay, S.: The structure of groups of prime power order, London Math. Soc. Monographs, New Series, p. 27. Oxford Univ. Press, Oxford (2002)
Liebeck, H.: Outer automorphisms in nilpotent \(p\)-groups of class \(2\). J Lond. Math. Soc. 40, 268–275 (1965)
Mazurov, V.D., Khukhro, E.I. (eds.): Unsolved problems in group theory, the kourovka notebook, No. 17. Russian Academy of Sciences, Siberian Division, Institute of Mathematics, Novosibirisk (2010)
Schmid, P.: A cohomological property for regular \(p\)-groups. Math. Z. 175, 1–3 (1980)
Shahriari, S.: On normal subgroups of capable groups. Arch. Math. 48, 193–198 (1987)
Acknowledgments
The first author would like to thank the Department of Mathematics at the University of the Basque Country for its excellent hospitality while part of this paper was being written; on the other hand, the first two authors also wish to thank Professors Norberto Gavioli, Gustavo A. Fernández Alcober and Carlo Maria Scoppola for their suggestions.
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Communicated by J. S. Wilson.
Marco Ruscitti would like to thank the Department of Mathematics at the University of the Basque Country for its excellent hospitality while part of this paper was being written. Leire Legarreta is supported by the Spanish Government, Grants MTM2011-28229-C02-02 and MTM2014-53810-C2-2-P, and by the Basque Government, Grant IT753-13. Manoj K. Yadav thanks INDAM and the Department of Mathematics at the University of L’Aquila for its excellent hospitality for a month during June–July, 2015.
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Ruscitti, M., Legarreta, L. & Yadav, M.K. Non-inner automorphisms of order p in finite p-groups of coclass 3. Monatsh Math 183, 679–697 (2017). https://doi.org/10.1007/s00605-016-0938-5
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DOI: https://doi.org/10.1007/s00605-016-0938-5