Classifying 2-groups by coclass

Newman, M.; O’Brien, E.

doi:10.1090/S0002-9947-99-02124-8

Classifying 2-groups by coclass
HTML articles powered by AMS MathViewer

by M. F. Newman and E. A. O’Brien

Trans. Amer. Math. Soc. 351 (1999), 131-169

DOI: https://doi.org/10.1090/S0002-9947-99-02124-8

PDF | Request permission

Abstract:

Now that the conjectures of Leedham-Green and Newman have been proved, we probe deeper into the classification of $p$-groups using coclass. We determine the pro-$2$-groups of coclass at most 3 and use these to classify the 2-groups of coclass at most 3 into families. Using extensive computational evidence, we make some detailed conjectures about the structure of these families. We also conjecture that the 2-groups of arbitrary fixed coclass exhibit similar behaviour.

References

N. Blackburn, On a special class of $p$-groups, Acta Math. 100 (1958), 45–92. MR 102558, DOI 10.1007/BF02559602
Wieb Bosma and John Cannon (1997), Handbook of Magma functions. School of Mathematics and Statistics, Sydney University.
Harold Brown, Rolf Bülow, Joachim Neubüser, Hans Wondratschek, and Hans Zassenhaus, Crystallographic groups of four-dimensional space, Wiley Monographs in Crystallography, Wiley-Interscience [John Wiley & Sons], New York-Chichester-Brisbane, 1978. MR 0484179
J. H. Conway, R. T. Curtis, S. P. Norton, R. A. Parker, and R. A. Wilson, $\Bbb {ATLAS}$ of finite groups, Oxford University Press, Eynsham, 1985. Maximal subgroups and ordinary characters for simple groups; With computational assistance from J. G. Thackray. MR 827219
Helga Finken (1979), “Raumgruppen und Serien von $p$-Gruppen”. Staatsexamensarbeit, RWTH Aachen.
B. Huppert, Endliche Gruppen. I, Die Grundlehren der mathematischen Wissenschaften, Band 134, Springer-Verlag, Berlin-New York, 1967 (German). MR 0224703, DOI 10.1007/978-3-642-64981-3
Bertram Huppert and Norman Blackburn, Finite groups. II, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 242, Springer-Verlag, Berlin-New York, 1982. AMD, 44. MR 650245
Rodney James, $2$-groups of almost maximal class, J. Austral. Math. Soc. 19 (1975), 343–357. MR 0382435, DOI 10.1017/S1446788700031554
Rodney James, Corrigendum: “$2$-groups of almost maximal class” [J. Austral. Math. Soc. 19 (1975), 343–357; MR 52 #3318], J. Austral. Math. Soc. Ser. A 35 (1983), no. 3, 307. MR 712807
C. R. Leedham-Green and M. F. Newman, Space groups and groups of prime-power order. I, Arch. Math. (Basel) 35 (1980), no. 3, 193–202. MR 583590, DOI 10.1007/BF01235338
C. R. Leedham-Green and Susan McKay, On the classification of $p$-groups of maximal class, Quart. J. Math. Oxford Ser. (2) 35 (1984), no. 139, 293–304. MR 755666, DOI 10.1093/qmath/35.3.293
C. R. Leedham-Green, S. McKay, and W. Plesken, Space groups and groups of prime-power order. V. A bound to the dimension of space groups with fixed coclass, Proc. London Math. Soc. (3) 52 (1986), no. 1, 73–94. MR 812446, DOI 10.1112/plms/s3-52.1.73
I. M. Isaacs, Recovering information about a group from its complex group algebra, Arch. Math. (Basel) 47 (1986), no. 4, 293–295. MR 866514, DOI 10.1007/BF01191352
C. R. Leedham-Green, The structure of finite $p$-groups, J. London Math. Soc. (2) 50 (1994), no. 1, 49–67. MR 1277754, DOI 10.1112/jlms/50.1.49
Roger C. Lyndon, Groups and geometry, London Mathematical Society Lecture Note Series, vol. 101, Cambridge University Press, Cambridge, 1985. Revised from “Groupes et géométrie” by A. Boidin, A. Fromageot and Lyndon. MR 794131, DOI 10.1017/CBO9781107325685
S. McKay, The precise bound to the coclass of space groups, J. London Math. Soc. (2) 50 (1994), no. 3, 488–500. MR 1299453, DOI 10.1112/jlms/50.3.488
M. F. Newman, Determination of groups of prime-power order, Group theory (Proc. Miniconf., Australian Nat. Univ., Canberra, 1975) Lecture Notes in Math., Vol. 573, Springer, Berlin, 1977, pp. 73–84. MR 0453862
M. F. Newman, Groups of prime-power order, Groups—Canberra 1989, Lecture Notes in Math., vol. 1456, Springer, Berlin, 1990, pp. 49–62. MR 1092222, DOI 10.1007/BFb0100730
M. F. Newman and E. A. O’Brien, Application of computers to questions like those of Burnside. II, Internat. J. Algebra Comput. 6 (1996), no. 5, 593–605. MR 1419133, DOI 10.1142/S0218196796000337
E. A. O’Brien, The $p$-group generation algorithm, J. Symbolic Comput. 9 (1990), no. 5-6, 677–698. Computational group theory, Part 1. MR 1075431, DOI 10.1016/S0747-7171(08)80082-X
E. A. O’Brien, Isomorphism testing for $p$-groups, J. Symbolic Comput. 16 (1993), no. 3, 305–320. MR 1259678, DOI 10.1006/jsco.1993.1049
Martin Schönert et al. (1997), GAP – Groups, Algorithms and Programming. Lehrstuhl D für Mathematik, RWTH Aachen.
Axel Schneider (1997), “Classifying $p$-groups by Coclass", M.Math. thesis, University of New South Wales.
C. R. Leedham-Green, The structure of finite $p$-groups, J. London Math. Soc. (2) 50 (1994), no. 1, 49–67. MR 1277754, DOI 10.1112/jlms/50.1.49
A. Shalev (1995), “Finite $p$-groups”, Finite and locally finite groups, NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci., 471, (Istanbul, 1994), pp. 401–450. Kluwer Acad. Publ., Dordrecht.
A. Shalev and E. I. Zel′manov, Pro-$p$ groups of finite coclass, Math. Proc. Cambridge Philos. Soc. 111 (1992), no. 3, 417–421. MR 1151320, DOI 10.1017/S0305004100075514
Charles C. Sims, Computation with finitely presented groups, Encyclopedia of Mathematics and its Applications, vol. 48, Cambridge University Press, Cambridge, 1994. MR 1267733, DOI 10.1017/CBO9780511574702