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Some questions about p-groups

Part of: Representation theory of groups

Published online by Cambridge University Press:  09 April 2009

Avinoam Mann
Avinoam Mann
Affiliation:
Einstein Institute of Mathematics Hebrew UniversityGivat Ram Jerusalem 91904Israel e-mail: mann@vms.huji.ac.il
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Abstract

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We present some questions that we feel are important and interesting for the theory of finite p-groups, and survey known results regarding these questions.

Mike, the other year you asked me how I saw the future of p-group theory. All I can offer in answer are more questions.

MSC classification

Secondary: 20D15: Nilpotent groups, $p$-groups
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 67 , Issue 3 , December 1999 , pp. 356 - 379
Copyright
Copyright © Australian Mathematical Society 1999

References

[Al]Alperin, J. L., ‘Large Abelian subgroups of p-groups’, Trans. Amer. Math. Soc. 117 (1965), 10–20.Google Scholar
[AG]Alperin, J. L. and Glauberman, G., ‘Limits of Abelian subgroups of finite p-groups’, J. Algebra 203 (1998), 533–566.CrossRefGoogle Scholar
[B1]Berkovich, Ja. G., ‘Subgroups and normal structure of a finite p-group’, Soviet Math. Dokl. 12 (1971), 71–75.Google Scholar
[B2]Berkovich, Y., ‘Groups of prime power order’, in: Finite groups (Berkovich, Y., Kazarin, L. and Zhmuď, E.) in preparation.Google Scholar
[Bi]Bialostocki, A., ‘On the other pα qβ theorem of Burnside’, Proc. Edinburgh Math. Soc. 30 (1987), 41–49.CrossRefGoogle Scholar
[BK]Bryant, R. M. and Kovács, L. G., ‘Lie representations and groups of prime power order’, J. London Math. Soc. 17 (1978), 415–421.CrossRefGoogle Scholar
[Bn]Blackburn, N., ‘On a special class of p-groups’, Acta Math. 100 (1958), 45–92.CrossRefGoogle Scholar
[Bs]Blackburn, S. R., ‘Groups of prime power order with derived subgroup of prime order’, Technical Report.Google Scholar
[BZ]Berkovich, Ya. G. and Zhmuď, E. M., Characters of finite groups 1 (Amer. Math. Soc., Providence, 1997).CrossRefGoogle Scholar
[C]Cartwright, M., ‘Class and breadth of a finite p-group’, Bull. London Math. Soc. 19 (1987), 425–430.CrossRefGoogle Scholar
[CH]Cossey, J. and Hawkes, T., ‘Sets of p-powers as conjugacy class sizes’, Proc. Amer. Math. Soc. to appear.Google Scholar
[CHM]Cossey, J., Hawkes, T. and Mann, A., ‘A criterion for a group to be nilpotent’, Bull. London Math. Soc. 24 (1992), 267–270.CrossRefGoogle Scholar
[Dd]Dade, E. C., ‘Products of orders of centralizers’, Math. Z. 96 (1967), 223–225.CrossRefGoogle Scholar
[Dv]Davitt, R. M., ‘The automorphism group of finite p-Abelian p-groups’, Illinois J. Math. 16 (1972), 76–85.CrossRefGoogle Scholar
[DDMS]Dixon, J. D., duSautoy, M. P. F., Mann, A. and Segal, D., Analytic pro-p groups, 2nd edition (Cambridge University Press), to appear.CrossRefGoogle Scholar
[dS]duSautoy, M. P. F., ‘Counting p-groups and nilpotent groups’, Technical Report.Google Scholar
[ERNS]Evans-Riley, S., Newman, M. F. and Schneider, C., ‘On the soluble length of groups with prime-power order’, Bull. Austral. Math. Soc. 59 (1999), 343–346.CrossRefGoogle Scholar
[Fa]Faudree, R., ‘A note on the automorphism group of a p-group’, Proc. Amer. Math. Soc. 19 (1968), 1379–1382.Google Scholar
[Fe]Feit, W., Characters of finite groups (Benjamin, New York, 1967).Google Scholar
[FS]Fernández-Alcober, G. A. and Shepherd, R. T., ‘On the order of p-groups of abundance zero’, J. Algebra 201 (1998), 392–400.CrossRefGoogle Scholar
[FT]Frame, J. S.Tamaschke, O., ‘Über die Ordnungen der Zentralisatoren der Elemente in endlichen Gruppen’, Math. Z. 83 (1964), 41–45.CrossRefGoogle Scholar
[Ga]Gaschütz, W., ‘Nichtablesche p-Gruppen besitzen äussere p-Automorphismen’, J. Algebra 4 (1966), 1–2.CrossRefGoogle Scholar
[Gi]Gillam, J. D., ‘A note on finite metabelian p-groups’, Proc. Amer. Soc. 25 (1970), 189–190.Google Scholar
[G11]Glauberman, G., ‘On Burnside's other pα qβ theorem’, Pacific J. Math. 56 (1975), 469–476.CrossRefGoogle Scholar
[G12]Glauberman, G., ‘Large Abelian subgroups of finite p-groups’, J. Algebra 196 (1997), 301–338.CrossRefGoogle Scholar
[H1]Higman, G., ‘Enumerating p-groups, I. Inequalitites’, Proc. London Math. Soc. (3) 10 (1960), 24–30.CrossRefGoogle Scholar
[H2]Higman, G., ‘Enumerating p-gruops, II. Preoblems whose solutions is PORC’, Proc. London Math. Soc. (3) 10 (1960), 566–582.CrossRefGoogle Scholar
[He]Heineken, H., ‘Nilpotente Gruppen, deren sämtliche Normalteiler charakteristich sind’, Arch. Math. 33 (1979/1980), 497–503.CrossRefGoogle Scholar
[Hi]Hilton, H., An introduction to the theory of groups of finite order (Oxford Press, Oxford, 1908).Google Scholar
[HKKP]Hassani, A., Khayaty, M., Khukhro, E. I. and Praeger, C. E., ‘Transitive permutation groups with bounded movement and maximal degree’, J. Algebra 214 (1999), 317–337.CrossRefGoogle Scholar
[HL]Heineken, H. and Liebeck, H., ‘The occurrence of finite groups in the automorphism group of nilpotent groups of class 2’, Arch. Math. 25 (1974), 8–16.CrossRefGoogle Scholar
[Ho]Hölder, O., ‘Die Gruppen the Ordnung p3, pq2, pqr, p4’, Math. Ann. 43 (1983), 301–412.CrossRefGoogle Scholar
[HP]Henn, H.-W. and Priddy, S., ‘p-nilpotence, classifying space indecomposability, and other properties of almost all finite groups’, Comment. Math. Helv. 69 (1994), 335–350.CrossRefGoogle Scholar
[Hu]Huppert, B., ‘A remark on the character-degrees of some p-groups’, Arch. Math. 59 (1992), 313–318.CrossRefGoogle Scholar
[Hy]Hyde, K. H., ‘On the order of the Sylow subgroups of the automorphism group of a finite group’, Glasgow Math. J. 11 (1970), 88–96.CrossRefGoogle Scholar
[I1]Isaacs, I. M., Character theory of finite groups (Academic Press, San Diego, 1992).Google Scholar
[I2]Isaacs, I. M., ‘Recovering information about a group form its complex group algebra’, Arch. Math. 47 (1986), 293–295.CrossRefGoogle Scholar
[I3]Isaacs, I. M., ‘Sets of p-powers as irreducible character degrees’, Proc. Amer. Math. Soc. 96 (1986), 551–552.Google Scholar
[I4]Isaacs, I. M., ‘Characters of groups associated with finite algebras’, J. Algebra 177 (1995), 708–730.CrossRefGoogle Scholar
[I5]Isaacs, I. M., ‘Groups with many equal classes’, Duke Math. J. 37 (1970), 501–506.CrossRefGoogle Scholar
[IK]Isaacs, I. M. and Karagueuzia, D., ‘conjugacy in groups of upper triangular matrices’, J. Albegra 202 (1998), 704–711.Google Scholar
[IP]Isaacs, I. M. and Passman, D. S., ‘A characterization of groups in terms of their irreducible character degrees’, Procific J. Math. 15 (1965), 899–903.Google Scholar
[JZ]Jaikin-Zapirain, A., ‘On the abundance of finite p-groups’, Technial Report.Google Scholar
[Ka]Kahn, B., ‘A characterisation of powerfully embedded normal subgroups of a p-group’, J. Algebra 188 (1997), 401–408.CrossRefGoogle Scholar
[Kh1]Khukhro, E. I., Nilpotent groups and their automorphisms (de Gruyter, Berlin, 1993).CrossRefGoogle Scholar
[Kh2]Khukhro, E. I., p-automorphisms of finite p-groups (Cambridge University Press, Cambridge, 1997).Google Scholar
[KJ]Konvisser, M. and Jonah, D., ‘Counting Abelian subgroups of p-groups, a projective approach’, J. Algebra 34 (1975), 309–330.CrossRefGoogle Scholar
[KLGP]Klaas, G., Leedham-Green, C. R. and Plesken, W., Linear pro-p groups of finite width, Lecture Notes in Math. 1674 (Springer, Berlin, 1997).CrossRefGoogle Scholar
[KN]Kirilov, A. A. and Neretin, Yu. A., ‘The variety An of n-dimensional Lie algebra structures’, Transl. Amer. Math. Soc. (2) 137 (1987), 21–30.Google Scholar
[Kr]Kirilov, A. A., ‘Variations on the triangular theme’, Transl. Amer. Math. Soc. (2) 169 (1995), 43–73.Google Scholar
[L]Lazard, M., ‘Sur les groupes nilpotents et les anneaux de Lie’, Ann. Sci. Ecole Norm. Sup. (3) 71 (1954), 101–190.CrossRefGoogle Scholar
[Lf]Laffey, T. J., ‘The minimum number of generators of a finite p-group’, Bull. London Math. soc. 5 (1973), 288–290.CrossRefGoogle Scholar
[LG]Leedham-Green, C. R., ‘The structure of finite p-goups’, J. London Math. Soc. 50 (1994), 49–67.CrossRefGoogle Scholar
[LGM]Leedham-Green, C. R. and McKay, S., ‘On the classfication of p-groups of maximal class’, Quart. J. Math. Oxford Ser. (2) 35 (1984), 293–304.CrossRefGoogle Scholar
[LGN]Leedham-Green, C. R. and Newman, M. F., ‘Space groups and groups of prime power order I’, Arch. Math. 35 (1980), 193–202.CrossRefGoogle Scholar
[LMM]Longobardi, P., Maj, M. and Mann, A., ‘Minimal classes and maximal class in p-groups’, Israel J. Math. 110 (1999), 93–102.CrossRefGoogle Scholar
[LMMW]Longobardi, P., MacHenry, T., Maj, M. and Wiegold, J., ‘On absolutely-nipotent of class k groups’, Rend. Mat. Acc. Lincei 6 (1995), 201–209.Google Scholar
[Mc]Macdonald, I. D., ‘Generalizations of a classical theorem about nilpotent groups’, Illinois J. Math. 8 (1964), 556–570.CrossRefGoogle Scholar
[Mi]Miller, G. A., ‘A non-Abelian group whose group of isomorphisms is Abelian’, Messenger Math. 43 (1913), 124–125Google Scholar
(or Miller, G. A., Collected works, vol. 5, 415–417).Google Scholar
[MM1]Mann, A. and Martinez, C., ‘The exponent of finite groups’, Arch. Math. 67 (1996), 8–10.CrossRefGoogle Scholar
[MM2]Mann, A. and Martinez, C., ‘Groups nearly of prime exponent and nearly Engel Lie algebras’, Arch. Math. 71 (1998), 5–11.CrossRefGoogle Scholar
[MN]McIver, A. and Neumann, P. M., ‘Enumerating finite groups’, Quart. J. Math. Oxford Ser. (2) 38 (1987), 473–488.CrossRefGoogle Scholar
[Mn1]Mann, A., ‘Enumerating finite groups and their defining relations’, J. Group Theory 1 (1998), 59–64.CrossRefGoogle Scholar
[Mn2]Mann, A., ‘Minimal characters of p-groups’, J. Group Theory 2 (1999), 225–250.CrossRefGoogle Scholar
[Mn3]Mann, A., ‘On p-groups whose maximal subgroups are isomorphic’, J. Austral. Math. Soc. (Ser. A) 59 (1995), 143–147.CrossRefGoogle Scholar
[Mn4]Mann, A., ‘The derived length of p-groups’, J. Algebra, to appear.Google Scholar
[Mn5]Mann, A., ‘Conjugacy classes in finite groups’, Israel J. Math. 31 (1978), 78–84.CrossRefGoogle Scholar
[Mn6]Mann, A., ‘The power structure of p-grups I’, J. Algebra 42 (1976), 121–135.CrossRefGoogle Scholar
[Mn7]Mann, A., ‘Enumerating finite groups and their defining relations II’, Technical Report.Google Scholar
[Mn8]Mann, A., ‘On the power strucuture of some p-groups’, in: Proceedings of the 2nd International Group Theory Conference, Supp. Rend. Circolo Mat. Palermo (2), vol. 23 (1990) pp. 227–235.Google Scholar
[MP]Mann, A. and Praeger, C. E., ‘Transitive permutation groups of minimal movement’, J. Algebra 181 (1996), 903–911.CrossRefGoogle Scholar
[Mr]Martin, U., ‘Almost all p-groups have automorphism group a p-group’, Bull. Amer. Math. Soc. 15 (1986), 78–82.CrossRefGoogle Scholar
[Mt]Mattarei, S., ‘On character tables of wreath products’, J. Albegra 175 (1995), 151–178.Google Scholar
[Nr]Neretin, Yu. A., ‘An estimate of the number of parameters defining an n-dimensional algebra’, Math. USSR-lzv. 30 (1988), 283–294.CrossRefGoogle Scholar
[Nw]Newman, M. F., ‘Groups of prime-power order’, in: Groups - Canberra 1989 (ed. Kovács, L. G.), Lecture Notes in Math. 1456 (Springer, Berlin, 1990), pp. 49–62.CrossRefGoogle Scholar
[NO1]Newman, M. F. and O'Brien, E. A., ‘A CAYLEY library for the groups of order dividing 128’, in: Group Theory, Proceedings of the Singapore conference (eds. Cheng, K. N. and Leong, Y. K.) (de Gruyter, Berlin, 1989) pp. 437–442.Google Scholar
[NO2]Newman, M. F. and O'Brien, E. A., ‘Classifying 2-groups by coclass’, Trans. Amer. Math. Soc. 351 (1999), 131–169.CrossRefGoogle Scholar
[NSW]Newmann, M. F., Sauerbier, G. and Wisliceny, J., ‘Groups of prime-power order with a small number of relations’, Rostock Math. Kolloq. 49 (1995), 141–154.Google Scholar
[P1]Pyber, L., ‘Enumberating finite groups of given order’, Ann. of Math. 137 (1993), 203–220.CrossRefGoogle Scholar
[P2]Pyber, L., ‘Group enumeration and where it leads us’, in: Proceedings of the European Congress of Mathematics (Budapest, 1996), to appear.Google Scholar
[P3]Pyber, L., ‘How Abelian if a finite group?’, in: The Mathematics of Paul Erdös (eds. Graham, R. L. and Nesetril, J.) (Springer, Berlin, 1998).Google Scholar
[PaS]Pálfy, P. P. and Szalay, M., ‘The distribution of the character degrees of the symmetric p-groups’, Acta Math. Hung. 41 (1983), 137–150.CrossRefGoogle Scholar
[PF]Posnick, F. (Fradkin), 3-Powerful groups and subgroups of powerful groups (Master's Thesis, Hebrew University, Jerusalem, 1998).Google Scholar
[PyS]Pyber, L. and Shalev, A., ‘Asymptotic results for primitive permutation groups’, J. Algebra 188 (1997), 103–124.CrossRefGoogle Scholar
[R]Ree, R., ‘The existence of outer automorphisms of some groups II’, Proc. Amer. Math. Soc. 9 (1958), 105–109.Google Scholar
[Sa]Sauerbier, G., ‘Zur Darstellung von Pro-2-Gruppen durch Erzeugende und Relationen’, Wiss. Z. Päd. Hochsch. Güstrow (Math. - Nat. Fak.) 1 (1986), 27–38.Google Scholar
[Sah]Sah, C. H., ‘Automorphisms of finite groups’, J. Algebra 10 (1968), 47–68.CrossRefGoogle Scholar
[Sch]Schmid, P., ‘Normal p-subgorups in the group of outer automorphisms of a finite p-group’, Math. Z. 147 (1976), 271–277.CrossRefGoogle Scholar
[Sh1]Shalev, A., ‘The structure of finite p-groups: effective proof of the coclass conjectures’, Invent. Math. 115 (1994), 315–345.CrossRefGoogle Scholar
[Sh2]Shalev, A., ‘Finite p-groups’, in: Finite and locally finite groups (Kluwer Acad. Publ., Dodrecht, 1995), pp. 401–450.CrossRefGoogle Scholar
[Sh3]Shalev, A., ‘Probabilistic group theory’, in: Groups St Andrews 1997 in Bath, II,, London Math. soc. Lecture Note Ser. 261 (Camrbridge University Press, Cambridge, 1999), pp. 648–678.CrossRefGoogle Scholar
[Si]Sims, C. C., ‘Enumerating p-groups’, Proc. London Math. Soc. 15 (1965), 151–166.CrossRefGoogle Scholar
[Sn]Sanders, P. J., ‘The coexponent of a regular p-group’, Comm. Algebra, to appear.Google Scholar
[Sp]shepherd, R. T., p-groups of maximal class (Ph.D. Thesis, University of Chicago, 1970).Google Scholar
[SW]Sanders, P. J. and Wilde, T. S., ‘The class and coexponent of a finite p-group’, Technical Report.Google Scholar
[T]Thomoson, J. G., ‘A non-duality theorem for finite groups’, J. Algebra 14 (1970), 1–4.CrossRefGoogle Scholar
[V]Verardi, L., ‘On groups whose non-central elements have the same finite number of conjugates’, Boll. Un. Mat. Ital. (7) 2-A (1988), 391–400.Google Scholar
[VA]Vera-López, A. and Arregi, J. M., ‘Some algorithms for the calculation of conjugacy classes in the Sylow p-subgroups of GL (n, q)’, J. Algebra 177 (1995), 899–925.CrossRefGoogle Scholar
[VLW]Vaughan-Lee, M. R. and Wiegold, J., ‘Generation of p-groups by elements of bounded breadth’, Proc. Royal. Soc. Edinburgh 95A (1983), 215–221.CrossRefGoogle Scholar
[Wb]Webb, U. H. M., ‘An elementary proof of Gaschütz' theorem’, Arch. Math. 35 (1980), 23–26.CrossRefGoogle Scholar
[Ws]Wisliceny, J., ‘Zur Darstellung von Pro-p-Gruppen und Lieschen Algebren durch Erzeugende und Relationen’, Math. Nachr. 102 (1981), 57–78.CrossRefGoogle Scholar
[Xu]Xu, M. Y., ‘Regular p-groups and their generalizations’, unpublished book manuscript.Google Scholar