A construction of the automorphism groups of indecomposable S-rings over \({Z_{2^n}}\)

Original Paper


Automorphism groups of indecomposable S-rings over cyclic groups \({Z_{2^n}}\) are considered. Such rings have been parameterized by atomic sequences, a collection of numerical data, introduced by the author (Lothar. Comb. 51 (Bb51h), 2005). In this paper an explicit construction of the groups is presented in terms of the atomic sequences.


S-ring Decomposable S-ring Automorphism group of an S-ring 

Mathematics Subject Classification (2000)

05E15 05E18 05E30 20B25 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. Evdokimov S.A., Ponomarenko I.N.: On a family of Schur rings over a finite cyclic group. St. Petersbg. Math. J. 13(3), 441–451 (2002)MATHMathSciNetGoogle Scholar
  2. Faradzhev I.A., Ivanov A.A., Klin M.H.: Galois correspondence between permutation groups and cellular rings (association schemes). Graphs Comb. 6, 303–332 (1990)CrossRefMATHMathSciNetGoogle Scholar
  3. Faradzhev, I.A., Klin, M.H., Muzychuk, M.E.: Cellular rings and groups of automorphisms of graphs. In: Faradzhev, I.A. et al. (eds.) Investigations on algebraic theory of combinatorial objects, mathematics and its applications, vol. 84, pp. 1–152. Kluwer, Dordrecht (1994)Google Scholar
  4. Gol’fand, J.J., Najmark, N.L., Pöschel, R.: The structure of S-rings over \({Z_{2^m}}\). Akad. der Wiss. der DDR, Inst. für Math., Preprint P-MATH-01/85, 1–30 (1985)Google Scholar
  5. Klin, M.H.: Automorphism groups of circulant graphs. Tagungsbericht of the conference: applicable algebra, Oberwolfach, 14–20 February (1993)Google Scholar
  6. Klin, M.H., Liskovets, V., Pöschel, R.: Analytical enumeration of circulant graphs with prime-squared number of vertices. Sèm. Lothar. Comb. 36 (B36d), 1–36 (1996)Google Scholar
  7. Klin, M.Ch., Najmark, N.L., Pöschel, R.: Schur rings over \({Z_{2^m}}\). Akad. der Wiss. der DDR Inst. für Math., Preprint P-MATH-14/81, 1–30 (1981)Google Scholar
  8. Klin, M.H., Pöschel, R.: The König Problem, the isomorphism problem for cyclic graphs and the method of Schur rings. In: Algebraic Methods in Graph Theory, Szeged, 1978, Colloq. Math. Soc. János Bolyai, vol. 25, pp. 405–434. North-Holland, Amsterdam (1981)Google Scholar
  9. Klin, M.H., Pöschel, R.: Circulant graphs via Schur ring theory. Automorphism groups of circulant graphs on p m vertices, p an odd prime (Manuscript)Google Scholar
  10. Kovács, I.: The number of indecomposable S-rings over a cyclic 2-group. Sèm. Lothar. Comb. 51 (Bb51h) (2005)Google Scholar
  11. Leung K.H., Ma S.L.: The structure of Schur rings over cyclic groups. J. Pure Appl. Algebra 66, 287–302 (1990)CrossRefMATHMathSciNetGoogle Scholar
  12. Liskovets V., Pöschel R.: Counting circulant graphs of prime-power order by decomposing into orbit enumeration problems. Discret. Math. 214, 173–191 (2000)CrossRefMATHGoogle Scholar
  13. Muzychuk M.E., Klin M.H., Pöschel R.: The isomorphism problem for circulant graphs via Schur ring theory. DIMACS Ser. Discret. Math. Theor. Comput. Sci. 56, 241–264 (2001)Google Scholar
  14. Pöschel R.: Untersuchungen von S-ringen, insbesondere im Gruppenring von p-Gruppen. Math. Nachr. 60, 1–27 (1974)CrossRefMATHMathSciNetGoogle Scholar
  15. Weisfeiler B.J.: On construction and identification of graphs. Lecture Notes in Mathematics. vol. 558. Springer, Berlin (1976)Google Scholar
  16. Wielandt H.: Finite Permutation Groups. Academic Press, New York (1964)MATHGoogle Scholar

Copyright information

© The Managing Editors 2011

Authors and Affiliations

  1. 1.FAMNIT, University of PrimorskaKoperSlovenia

Personalised recommendations