Home A necessary condition for the Smith equivalence of G-modules and its sufficiency
Article
Licensed
Unlicensed Requires Authentication

A necessary condition for the Smith equivalence of G-modules and its sufficiency

  • Masaharu Morimoto EMAIL logo
Published/Copyright: November 3, 2016
Become an author with De Gruyter Brill
Author Information Explore this Subject
Mathematica Slovaca
From the journal Mathematica Slovaca Volume 66 Issue 4

Abstract

Let G be a finite group. In this paper we give a new necessary condition for two real G-modules to be Smith equivalent if G has a normal Sylow 2-subgroup. We show that the condition is also sufficient under certain hypotheses. By results on the Smith equivalence obtained in this paper, the primary Smith sets are not subgroups of the real representation rings for various Oliver groups with normal Sylow 2-subgroups.

MSC 2010: Primary 57S17; Secondary 20C15
keywords: Smith equivalence; Smith set; fixed point; representation

(Communicated by Július Korbaš)

Acknowledgements

The author would like to thank Július Korbaš and the referee for their carefully reading a draft and giving valuable comments.

References

[1] Bierstone, E.: General position of equivariant maps, Trans. Amer. Math. Soc. 234 (1977), 447ߞ466.10.1090/S0002-9947-1977-0464287-3Search in Google Scholar

[2] Bredon, G. E.: Topology and Geometry. Grad. Texts in Math. 139, Springer-Verlag, New York, 1993.10.1007/978-1-4757-6848-0Search in Google Scholar

[3] Cappell, S. E.—Shaneson, J. L.: Representations at fixed points. In: Group Actions on Manifolds, Boulder, Colo., 1983 (R. Schultz, ed.). Contemp. Math. 36, Amer. Math. Soc., Providence, RI, 1985, pp. 151ߞ158.10.1090/conm/036/780963Search in Google Scholar

[4] Cohen, M. M.: A Course in Simple-Homotopy Theory. Grad. Texts in Math. 10, Springer-Verlag, New York, 1973.10.1007/978-1-4684-9372-6Search in Google Scholar

[5] Conner, P. E.—Floyd, E. E.: Differentiable Periodic Maps. Ergeb. Math. Grenzgeb. (N.F) 33, Springer-Verlag, Berlin-G—ottingen-Heiderberg, 1964.10.1007/978-3-662-34580-1Search in Google Scholar

[6] Dieck, T.: Transformation Groups, de Gruyter Stud. Math. 8, Walter de Gruyter & Co., Berlin, 198710.1515/9783110858372Search in Google Scholar

[7] Edmonds, A. L.—Lee, R.: Fixed point sets of group actions on Euclidean space, Topology 14 (1975), 339ߞ345.10.1016/0040-9383(75)90018-XSearch in Google Scholar

[8] Hattori, A.: Manifolds. Iwanami Zensho, No. 288, Iwanami, Tokyo, 1976 (Japanese).Search in Google Scholar

[9] Hirsch, M. W.: Differential Topology, Grad. Texts in Math. 33, Springer-Verlag, New York, 1976.10.1007/978-1-4684-9449-5Search in Google Scholar

[10] Ju, X. M.: The Smith set of the group S5 — C2 — × × C2, Osaka J. Math. 47 (2010), 215ߞ236.Search in Google Scholar

[11] Koto, A.—Morimoto, M.—QI, Y.: The Smith sets of finite groups with normal Sylow 2-subgroups and small nilquotients, J. Math. Kyoto Univ. 48 (2008), 219ߞ22710.1215/kjm/1250280981Search in Google Scholar

[12] Laitinen, E.—Morimoto, M.: Finite groups with smooth one fixed point actions on spheres, ForumMath. 10 (1998), 479ߞ520.10.1515/form.10.4.479Search in Google Scholar

[13] Laitinen, E.—Morimoto, M.—pawałowski, K.: Deleting-inserting theorem for smooth actions of finite nonsolvable groups on spheres, Comment. Math. Helv. 70 (1995), 10ߞ38.10.1007/BF02565998Search in Google Scholar

[14] Laitinen, E.—Pawałowski, K.: Smith equivalence of representations for finite perfect groups, Proc.Amer. Math. Soc. 127 (1999), 297ߞ307.10.1090/S0002-9939-99-04544-XSearch in Google Scholar

[15] Morimoto, M.: Smith equivalent Aut(A6)-representations are isomorphic, Proc. Amer. Math. Soc. 136 (2008), 3683ߞ3688.10.1090/S0002-9939-08-08891-6Search in Google Scholar

[16] Morimoto, M.: Fixed-point sets of smooth actions on spheres, J. K-Theory 1 (2008), 95ߞ128.10.1017/is007011012jkt009Search in Google Scholar

[17] Morimoto, M.: Nontrivial P(G)-matched S-related pairs for finite gap Oliver groups, J. Math. Soc. Japan 62 (2010), 623ߞ647.10.2969/jmsj/06220623Search in Google Scholar

[18] Morimoto, M.: Deleting and inserting fixed point manifolds under the weak gap condition, Publ. Res. Inst. Math. Sci. 48 (2012), 623ߞ651.10.2977/PRIMS/84Search in Google Scholar

[19] Morimoto, M.: Tangential representations of one-fixed-point actions on spheres and Smith equivalence, J. Math. Soc. Japan 67 (2015), 195ߞ205.10.2969/jmsj/06710195Search in Google Scholar

[20] Morimoto, M.—Qi, Y.: Study of the Smith sets of gap Oliver groups. In: RIMS Kôkyûroku Bessatsu 1670,Res. Inst. Math. Sci. (RIMS), Kyoto, 2009, pp. 126ߞ139.Search in Google Scholar

[21] Morimoto, M.—Qi, Y.: The primary Smith sets of finite Oliver groups. In: Group Actions and Homogeneous Spaces, Bratislava, 2009 (J. Korbaš, M. Morimoto, K. Pawałowski, eds.), Fak. Mat. Fyz. Inform. Univ.Komenskeho, Bratislava, 2010, pp. 61ߞ73. http://www.fmph.uniba.sk/index.php?id=2796Search in Google Scholar

[22] Morimoto, M.—YANAGIHARA, M.: The gap condition for S5 and GAP programs, J. Fac. Env. Sci. Tech.Okayama Univ. 1 (1996), 1ߞ13.Search in Google Scholar

[23] Oliver, B.: Fixed point sets and tangent bundles of actions on disks and Euclidean spaces, Topology 35 (1996), 583ߞ615.10.1016/0040-9383(95)00043-7Search in Google Scholar

[24] Oliver, R.: Fixed-point sets of group actions on finite acyclic complexes, Comment. Math. Helv. 50 (1975), 155ߞ177.10.1007/BF02565743Search in Google Scholar

[25] Pawałowski, K.—Solomon, R.: Smith equivalence and finite Oliver groups with Laitinen number 0 or 1 Algebr. Geom. Topol. 2 (2002), 843ߞ895.10.2140/agt.2002.2.843Search in Google Scholar

[26] Pawałowski, K.—Sumi, S.: The Laitinen Conjecture for finite non-solvable groups, Proc. Edinb. Math. Soc. (2) 56 (2013), 303ߞ336.10.1017/S0013091512000223Search in Google Scholar

[27] Petrie, T.: Pseudoequivalences of G-manifolds. In: Algebraic and Geometric Topology, Stanford, Calif., 1976, Part 1. Proc. Sympos. Pure Math. XXXII, Amer. Math. Soc., Providence, RI, 1978, pp. 169ߞ210.10.1090/pspum/032.1/520505Search in Google Scholar

[28] Petrie, T.: Three theorems in transformation groups. In: Algebraic Topology, Aarhus, 1978 (J. Dupont, I. Madsen, eds.). Lecture Notes in Math. 763, Springer-Verlag, Berlin-Heidelberg-New York, 1979, pp. 549ߞ572.10.1007/BFb0088102Search in Google Scholar

[29] Petrie, T.—Randall, J. D.: Transformation Groups on Manifolds Monographs and Textbooks in Pure and Applied Mathematics, No. 82, Marcel Dekker, Inc., New York-Basel, 1984.Search in Google Scholar

[30] Sanchez, C. U.: Actions of groups of odd order on compact, orientable manifolds, Proc. Amer. Math. Soc. 54 (1976), 445ߞ448.10.1090/S0002-9939-1976-0407871-XSearch in Google Scholar

[31] Smith, P. A.: New results and old problems in finite transformation groups, Bull. Amer. Math. Soc. 66 (1960), 401ߞ415.10.1090/S0002-9904-1960-10491-0Search in Google Scholar

[32] Steenrod, N. E.: The Topology of Fibre Bundles, Princeton University Press, Princeton, NJ, 1951.10.1515/9781400883875Search in Google Scholar

[33] Sumi, T.: Smith set for a nongap Oliver group. In: RIMS Kôkyûroku Bessatsu 1670, Res. Inst. Math. Sci.(RIMS), Kyoto, 2009, pp. 25ߞ33.Search in Google Scholar

[34] Sumi, T.: Smith set for a simple group, In: RIMS Kôkyûroku Bessatsu 1732, Res. Inst. Math. Sci. (RIMS), Kyoto, 2011, pp. 101ߞ106 (Japanese).Search in Google Scholar

[35] Sumi, T.: The gap hypothesis for finite groups which have an abelian quotient group not of order a power of 2, J. Math. Soc. Japan 64 (2012), 91ߞ106.10.2969/jmsj/06410091Search in Google Scholar

[36] Wasserman, A.: Equivariant differential topology, Topology 8 (1969), 127ߞ150.10.1016/0040-9383(69)90005-6Search in Google Scholar

Received: 2013-12-3
Accepted: 2014-3-23
Published Online: 2016-11-3
Published in Print: 2016-8-1

© 2016 Mathematical Institute Slovak Academy of Sciences

Articles in the same Issue

  1. Research Article
  2. Effect algebras with a full set of states
  3. Research Article
  4. An approach to orthomodular lattices via lattices with an antitone involution
  5. Research Article
  6. On properties of the free BCI-algebra with one generator
  7. Research Article
  8. The generalized order-k Narayana's cows numbers
  9. Research Article
  10. The irrationality of infinite series of a special kind
  11. Research Article
  12. On skew-commuting mappings in semiprime rings
  13. Research Article
  14. On the subordination and superordination of strongly starlike functions
  15. Research Article
  16. On a subclass of analytic functions involving Sălăgean integral operator
  17. Research Article
  18. Uniqueness of meromorphic functions sharing a value or small function
  19. Research Article
  20. A new inclusion for Bavrin's families of holomorphic functions in n-circular domains
  21. Research Article
  22. Oscillation of nonlinear fourth order mixed neutral differential equations
  23. Research Article
  24. Oscillation results for third order nonlinear mixed neutral differential equations
  25. Research Article
  26. Homoclinic solutions for second order Hamiltonian systems with general potentials
  27. Research Article
  28. A class of fractional impulsive functional differential equations with nonlocal conditions
  29. Research Article
  30. The weighted reverse poincaré type inequality for the difference of two parabolic subsolutions
  31. Research Article
  32. On the well posed solutions for nonlinear second order neutral difference equations
  33. Research Article
  34. Some approximation results for operators of Szász-Mirakjan-Durrmeyer type
  35. Research Article
  36. On the geometry of conditional expectations treated as projections on the L2-space
  37. Research Article
  38. Spectral problems of nonself-adjoint singular discrete Sturm-Liouville operators
  39. Research Article
  40. A necessary condition for the Smith equivalence of G-modules and its sufficiency
  41. Research Article
  42. On the internal approach to differential equations. 1. The involutiveness and standard basis
  43. Research Article
  44. Law of inertia for the factorization of cubic polynomials — the case of discriminants divisible by three
https://doi.org/10.1515/ms-2015-0197
Keywords for this article
Smith equivalence; Smith set; fixed point; representation

Articles in the same Issue

  1. Research Article
  2. Effect algebras with a full set of states
  3. Research Article
  4. An approach to orthomodular lattices via lattices with an antitone involution
  5. Research Article
  6. On properties of the free BCI-algebra with one generator
  7. Research Article
  8. The generalized order-k Narayana's cows numbers
  9. Research Article
  10. The irrationality of infinite series of a special kind
  11. Research Article
  12. On skew-commuting mappings in semiprime rings
  13. Research Article
  14. On the subordination and superordination of strongly starlike functions
  15. Research Article
  16. On a subclass of analytic functions involving Sălăgean integral operator
  17. Research Article
  18. Uniqueness of meromorphic functions sharing a value or small function
  19. Research Article
  20. A new inclusion for Bavrin's families of holomorphic functions in n-circular domains
  21. Research Article
  22. Oscillation of nonlinear fourth order mixed neutral differential equations
  23. Research Article
  24. Oscillation results for third order nonlinear mixed neutral differential equations
  25. Research Article
  26. Homoclinic solutions for second order Hamiltonian systems with general potentials
  27. Research Article
  28. A class of fractional impulsive functional differential equations with nonlocal conditions
  29. Research Article
  30. The weighted reverse poincaré type inequality for the difference of two parabolic subsolutions
  31. Research Article
  32. On the well posed solutions for nonlinear second order neutral difference equations
  33. Research Article
  34. Some approximation results for operators of Szász-Mirakjan-Durrmeyer type
  35. Research Article
  36. On the geometry of conditional expectations treated as projections on the L2-space
  37. Research Article
  38. Spectral problems of nonself-adjoint singular discrete Sturm-Liouville operators
  39. Research Article
  40. A necessary condition for the Smith equivalence of G-modules and its sufficiency
  41. Research Article
  42. On the internal approach to differential equations. 1. The involutiveness and standard basis
  43. Research Article
  44. Law of inertia for the factorization of cubic polynomials — the case of discriminants divisible by three
Sign up now to receive a 20% welcome discount
Institutional Access
How does access work?
Have an idea on how to improve our website?
Please write us.
© 2025 De Gruyter Brill
Downloaded on 26.9.2025 from https://www.degruyterbrill.com/document/doi/10.1515/ms-2015-0197/html
Scroll to top button