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FINITE $p$-GROUPS WITH SMALL AUTOMORPHISM GROUP

Part of: Representation theory of groups

Published online by Cambridge University Press:  20 April 2015

JON GONZÁLEZ-SÁNCHEZ  and
ANDREI JAIKIN-ZAPIRAIN
JON GONZÁLEZ-SÁNCHEZ
Affiliation:
Departamento de Matemáticas, Facultad de Ciencia y Tecnología, Universidad del País Vasco-Euskal Herriko Unibertsitatea, Apartado 644 48080 Bilbao, Spain
ANDREI JAIKIN-ZAPIRAIN
Affiliation:
Departamento de Matemáticas, Universidad Autónoma de Madrid, and Instituto de Ciencias Matemáticas, CSIC-UAM-UC3M-UCM, 28049-Madrid, Spain; andrei.jaikin@uam.es

Abstract

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For each prime $p$ we construct a family $\{G_{i}\}$ of finite $p$-groups such that $|\text{Aut}(G_{i})|/|G_{i}|$ tends to zero as $i$ tends to infinity. This disproves a well-known conjecture that $|G|$ divides $|\text{Aut}(G)|$ for every nonabelian finite $p$-group $G$.

MSC classification

Primary: 20D15: Nilpotent groups, $p$-groups
Secondary: 20D45: Automorphisms
Type
Research Article
Information
Forum of Mathematics, Sigma , Volume 3 , 2015 , e7
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Creative Commons
Creative Common License - CCCreative Common License - BY
This is an Open Access article, distributed under the terms of the Creative Commons Attribution licence (http://creativecommons.org/licenses/by/3.0/), which permits unrestricted re-use, distribution, and reproduction in any medium, provided the original work is properly cited.
Copyright
© The Author(s) 2015

References

Bray, J. N. and Wilson, R. A., ‘On the orders of automorphism groups of finite groups’, Bull. Lond. Math. Soc. 37 (2005), 381–385.Google Scholar
Bray, J. N. and Wilson, R. A., ‘On the orders of automorphism groups of finite groups II’, J. Group Theory 9 (2006), 537–545.Google Scholar
Buckley, J., ‘Automorphism groups of isoclinic p-groups’, J. Lond. Math. Soc. (2) 12 (1975/76), 37–44.Google Scholar
Davitt, R. M., ‘The automorphism group of a finite metacyclic p-group’, Proc. Amer. Math. Soc. 25 (1970), 876–879.Google Scholar
Davitt, R. M., ‘The automorphism group of finite p-abelian p-groups’, Illinois J. Math. 16 (1972), 76–85.Google Scholar
Davitt, R. M., ‘On the automorphism group of a finite p-group with a small central quotient’, Canad. J. Math. 32 (1980), 1168–1176.CrossRefGoogle Scholar
Davitt, R. M. and Otto, A. D., ‘On the automorphism group of a finite p-group with the central quotient metacyclic’, Proc. Amer. Math. Soc. 30 (1971), 467–472.Google Scholar
Davitt, R. M. and Otto, A. D., ‘On the automorphism group of a finite modular p-group’, Proc. Amer. Math. Soc. 35 (1972), 399–404.Google Scholar
Dixon, J., du Sautoy, M., Mann, A. and Segal, D., Analytic Pro-p Groups, 2nd edn (Cambridge University Press, Cambridge, 1999).CrossRefGoogle Scholar
Eick, B., ‘Automorphism groups of 2-groups’, J. Algebra 300 (2006), 91–101.Google Scholar
Exarchakos, T., ‘LA-groups’, J. Math. Soc. Japan 33 (1981), 185–190.CrossRefGoogle Scholar
Exarchakos, T., ‘On p-groups of small order’, Publ. Inst. Math. (Beograd) (N.S.) 45(59) (1989), 73–76.Google Scholar
Faudree, R., ‘A note on the automorphism group of a p-group’, Proc. Amer. Math. Soc. 19 (1968), 1379–1382.Google Scholar
Fouladi, S., Jamali, A. R. and Orfi, R., ‘Automorphism groups of finite p-groups of coclass 2’, J. Group Theory 10 (2007), 437–440.CrossRefGoogle Scholar
Gaschütz, W., ‘Kohomologische Trivialitäten und äussere Automorphismen von p-Gruppen’, Math. Z. 88 (1965), 432–433.Google Scholar
Gavioli, N., ‘The number of automorphisms of groups of order p 7’, Proc. R. Irish Acad. Sect. A 93 (1993), 177–184.Google Scholar
Green, J. A., ‘On the number of automorphisms of a finite group’, Proc. R. Soc. Lond. A 237 (1956), 574–581.Google Scholar
Hummel, K. G., ‘The order of the automorphism group of a central product’, Proc. Amer. Math. Soc. 47 (1975), 37–40.CrossRefGoogle Scholar
Hyde, K. H., ‘On the order of the Sylow subgroups of the automorphism group of a finite group’, Glasg. Math. J. 11 (1970), 88–96.Google Scholar
Lazard, M., ‘Groupes analytiques p-adiques’, Publ. Math. Inst. Hautes Études Sci. 26 (1965), 389–603.Google Scholar
Ledermann, W. and Neumann, B. H., ‘On the order of the automorphism group of a finite group I’, Proc. R. Soc. Lond. A 233 (1956), 494–506.Google Scholar
Luks, E., ‘Lie algebras with only inner derivations need not be complete’, J. Algebra 15 (1970), 280–282.Google Scholar
Mazurov, V. D. and Khukhro, E. I., The Kourovka Notebook. Unsolved Problems in Group Theory, 17th augmented edn (Russian Academy of Sciences Siberian Division, Institute of Mathematics, 2010).Google Scholar
Neukirch, J., Schmidt, A. and Wingberg, K., Cohomology of Number Fields, 2nd edn, Grundlehren der Mathematischen Wissenschaften, 323 (Springer, Berlin, 2008).Google Scholar
Otto, A. D., ‘Central automorphisms of a finite p-group’, Trans. Amer. Math. Soc. 125 (1966), 280–287.Google Scholar
Ree, R., ‘The existence of outer automorphisms of some groups II’, Proc. Amer. Math. Soc. 9 (1958), 105–109.Google Scholar
Sato, T., ‘The derivations of the Lie algebras’, Tôhoku Math. J. (2) 23 (1971), 21–36.Google Scholar
Schenkman, E., ‘The existence of outer automorphisms of some nilpotent groups of class 2’, Proc. Amer. Math. Soc. 6 (1955), 6–11.Google Scholar
Symonds, P. and Weigel, T., ‘Cohomology of p-adic analytic groups’, inNew Horizons in Pro-p Groups, Progress in Mathematics, 184 (Birkhäuser Boston, Boston, MA, 2000), 349–410.Google Scholar
Thillaisundaram, A., ‘The automorphism group for p-central p-groups’, Int. J. Group Theory 1 (2012), 59–71.Google Scholar
Yadav, M. K., ‘On automorphisms of finite p-groups’, J. Group Theory 10 (2007), 859–866.Google Scholar