Skip to main content
SpringerLink
Log in

On the Structure of Finite Groups with Periodic Cohomology

  • Chapter
  • First Online:
Lie Groups: Structure, Actions, and Representations

Part of the book series: Progress in Mathematics ((PM,volume 306))

Abstract

Detailed analysis of the structure of groups with periodic cohomology, with application to the problem of free group actions on spheres.

Dedicated to Joe Wolf in celebration of his 75th birthday

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 84.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 109.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 109.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Notes

  1. 1.

    Incorrectly given in [41] as 3.

References

  1. H. Bender, The Brauer–Suzuki–Wall theorem, Illinois J. Math. 19 (1974), 229–235.

    Google Scholar 

  2. H. Bender, Finite groups with dihedral Sylow 2-subgroups, J. Algebra 70 (1981), 216–228.

    Google Scholar 

  3. S. Bentzen and I. Madsen, On the Swan subgroup of certain periodic groups. Math. Ann. 264 (1983), 447–474.

    Google Scholar 

  4. S. Bentzen, Some numerical results on space forms, Proc. London Math. Soc. 54 (1987), 559–576.

    Google Scholar 

  5. R. Brauer, M. Suzuki and G. E. Wall, A characterization of the one-dimensional unimodular groups over finite fields, Illinois J. Math. 2 (1958), 718–745.

    Google Scholar 

  6. W. Browder, T. Petrie and C. T. C. Wall, The classification of free actions of cyclic groups of odd order on homotopy spheres, Bull. Amer. Math. Soc. 77 (1971), 455–459.

    Google Scholar 

  7. G. Brumfiel, I. Madsen and J. Milgram, PL characteristic classes and cobordism, Ann. of Math. 97 (1973), 82–159.

    Google Scholar 

  8. W. Burnside, On finite groups in which all the Sylow subgroups are cyclical, Messenger of Math. 35 (1905), 46–50.

    Google Scholar 

  9. W. Burnside, Theory of Groups of Finite Order (2nd. ed.) Cambridge Univ. Press, 1911; (Reprinted by Dover).

    Google Scholar 

  10. H. Cartan and S. Eilenberg, Homological Algebra, Oxford Univ. Press, 1956.

    MATH  Google Scholar 

  11. J. F. Davis and J. Milgram, Semicharacteristics, bordism, and free group actions. Trans. Amer. Math. Soc. 312 (1989), 55–83; (1987), 559–576.

    Google Scholar 

  12. A. Fröhlich, M. E. Keating and S. M. J. Wilson, The class groups of quaternion and dihedral 2-groups, Mathematika 21 (1974), 64–71.

    Google Scholar 

  13. S. Galovitch, I. Reiner and S. Ullom, Class groups for integral representations of metacyclic groups, Mathematika 19 (1972), 105–111.

    Google Scholar 

  14. G. Glauberman, On groups with a quaternion Sylow 2-subgroup, Illinois J. Math. 18 (1974), 60–65.

    Google Scholar 

  15. D. Gorenstein, Finite Groups, Harper & Row, 1968.

    Google Scholar 

  16. D. Gorenstein and J. H. Walter, The characterisation of finite groups with dihedral Sylow 2-subgroups, J. Alg. 2 (1965), 85–151, 218–270, 334–393.

    Google Scholar 

  17. I. Hambleton and I. Madsen, Actions of finite groups on R n + k with fixed set R k, Canad. J. Math. 38 (1986), 781–860.

    Google Scholar 

  18. I. Hambleton and I. Madsen, Local surgery obstructions and space forms, Math. Z. 193 (1986), 191–214.

    Google Scholar 

  19. I. Hambleton, R. J. Milgram, L. Taylor & B. Williams, Surgery with finite fundamental group, Proc. London Math. Soc. 56 (1988), 349–379.

    Google Scholar 

  20. E. Laitinen and I. Madsen, Topological classifications of \(\mathrm{Sl}_{2}(F_{p})\) space forms, Algebraic Topology, Aarhus 1978, Lecture Notes in Math., 763, Springer, 1979, pp. 235–261.

    Google Scholar 

  21. E. Laitinen and I. Madsen, The L theory of groups with periodic cohomology, preprint, Aarhus, 1981.

    Google Scholar 

  22. T. Y. Lam, Artin exponent of finite groups, J. Algebra 9 (1968), 94–119.

    Google Scholar 

  23. R. Lee, Semicharacteristic classes, Topology 12 (1973), 183–199.

    Google Scholar 

  24. T. Macko and C. Wegner, On fake lens spaces with the fundamental group of order a power of 2, Alg. Geom. Top. 9 (2009), 1837–1883.

    Google Scholar 

  25. T. Macko and C. Wegner, On the classification of fake lens spaces, Forum Math. 23 (2011), 1053–1091.

    Google Scholar 

  26. S. MacLane, Homology, Springer-Verlag, 1963.

    Google Scholar 

  27. I. Madsen, Reidemeister torsion, surgery invariants and spherical space forms. Proc. London Math. Soc. (3) 46 (1983), 193–240.

    Google Scholar 

  28. I. Madsen and R. J. Milgram, The classifying spaces for surgery and cobordism of manifolds, Ann. of Math. Study 92, Princeton Univ. Press, 1979.

    Google Scholar 

  29. I. Madsen, C. B. Thomas and C. T. C. Wall, The topological spherical space-form problem II, Existence of free actions, Topology 15 (1976), 375–382.

    Google Scholar 

  30. I. Madsen, C. B. Thomas and C. T. C. Wall, The topological spherical space-form problem III, Dimensional bounds and smoothing, Pacific. J. Math. 106 (1983), 135–143.

    Google Scholar 

  31. R. J. Milgram, A survey of the compact space form problem, Symposium on Algebraic Topology in honor of José Adem (Oaxtepec, 1981), Contemp. Math. 12, Amer. Math. Soc., 1982, pp. 219–255.

    Google Scholar 

  32. R. J. Milgram, Evaluating the Swan finiteness obstruction for periodic groups, Algebraic and geometric topology (New Brunswick, N.J., 1983), Lecture Notes in Math., 1126, Springer, 1985, pp 127–158.

    Google Scholar 

  33. J. W. Milnor, Groups which act on S n without fixed points, Amer. J. Math. 79 (1957), 632–630.

    Google Scholar 

  34. R. Oliver, Projective class groups of integral group rings: a survey, Orders and their applications (Oberwolfach, 1984), Lecture Notes in Math. 1142, Springer, 1985, pp 211–232.

    Google Scholar 

  35. R. Oliver, Whitehead Groups of Finite Groups, London Math. Soc. Lecture Notes 132, Cambridge Univ. Press, 1988. viii+349 pp.

    Google Scholar 

  36. P. A. Smith, Permutable periodic transformations, Proc. Nat. Acad. Sci. U. S. A. 30 (1944), 105–108.

    Google Scholar 

  37. M. Suzuki, On finite groups with cyclic Sylow subgroups for all odd primes, Amer. J. Math. 77 (1955), 657–691.

    Google Scholar 

  38. R. G. Swan, Induced representations and projective modules, Ann. of Math. 71 (1960), 552–578.

    Google Scholar 

  39. R. G. Swan, The p-period of a finite group, Illinois J. Math. 4 (1960), 341–346.

    Google Scholar 

  40. R. G. Swan, Periodic resolutions for finite groups, Ann. of Math. 72 (1960), 267–291.

    Google Scholar 

  41. C. B. Thomas and C. T. C. Wall, The topological spherical space-form problem I, Comp. Math. 23 (1971), 101–114.

    Google Scholar 

  42. C. T. C. Wall, Surgery on Compact Manifolds (London Math. Soc. monographs no. 1) x, 280 pp. Academic Press, 1970. Second edition, with editorial comments by A. A. Ranicki, xvi, 302 pp. Amer. Math. Soc. 1999.

    Google Scholar 

  43. C. T. C. Wall, Norms of units in group rings, Proc. London Math. Soc. 29 (1974), 593–632.

    Google Scholar 

  44. C. T. C. Wall, Classification of Hermitian forms VI: group rings, Ann. of Math. 103 (1976), 1–80.

    Article  MathSciNet  MATH  Google Scholar 

  45. C. T. C. Wall, Free actions of finite groups on spheres, Proc. Symp. in Pure Math. 32i (Algebraic and Geometric topology) (ed. J. Milgram) Amer. Math. Soc. 1978, pp 115–124.

    Google Scholar 

  46. C. T. C. Wall, Periodic projective resolutions, Proc. London Math. Soc. 39 (1979), 509–533.

    Google Scholar 

  47. J. A. Wolf, Spaces of Constant Curvature, McGraw-Hill, 1967.

    Google Scholar 

  48. H. Zassenhaus, Uber endliche Fastkörper, Abh. Math. Sem. Hamburg 11 (1936), 187–220.

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematical Sciences, University of Liverpool, Liverpool, Merseyside, L69 3BX, UK

    C. T. C. Wall

Authors
  1. C. T. C. Wall
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to C. T. C. Wall .

Editor information

Editors and Affiliations

  1. Remagen, Germany

    Alan Huckleberry

  2. Jacobs University School of Engineering and Science, Bremen, Germany

    Ivan Penkov

  3. Yale University Dept. Mathematics, New Haven, Connecticut, USA

    Gregg Zuckerman

Rights and permissions

Reprints and permissions

Copyright information

© 2013 Springer Science+Business Media New York

About this chapter

Cite this chapter

Wall, C.T.C. (2013). On the Structure of Finite Groups with Periodic Cohomology. In: Huckleberry, A., Penkov, I., Zuckerman, G. (eds) Lie Groups: Structure, Actions, and Representations. Progress in Mathematics, vol 306. Birkhäuser, New York, NY. https://doi.org/10.1007/978-1-4614-7193-6_16

Download citation

Publish with us

Policies and ethics

Search

Navigation