Central European Journal of Mathematics

, Volume 10, Issue 2, pp 722–739 | Cite as

Schreier type theorems for bicrossed products

Research Article
  • 41 Downloads

Abstract

We prove that the bicrossed product of two groups is a quotient of the pushout of two semidirect products. A matched pair of groups (H;G; α; β) is deformed using a combinatorial datum (σ; v; r) consisting of an automorphism σ of H, a permutation v of the set G and a transition map r: GH in order to obtain a new matched pair (H; (G; *); α′, β′) such that there exists a σ-invariant isomorphism of groups H α⋈β GH α′⋈β′ (G, *). Moreover, if we fix the group H and the automorphism σ ∈ Aut H then any σ-invariant isomorphism H α⋈β GH α′⋈β′ G′ between two arbitrary bicrossed product of groups is obtained in a unique way by the above deformation method. As applications two Schreier type classification theorems for bicrossed products of groups are given.

Keywords

Matched pairs Bicrossed product of groups 

MSC

20B05 20B35 20D06 20D40 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [1]
    Agore A.L., Chirvăsitu A., Ion B., Militaru G., Bicrossed products for finite groups, Algebr. Represent. Theory, 2009, 12(2–5), 481–488MathSciNetMATHCrossRefGoogle Scholar
  2. [2]
    Aguiar M., Andruskiewitsch N., Representations of matched pairs of groupoids and applications to weak Hopf algebras, In: Algebraic Structures and their Representations, Contemp. Math., 376, American Mathematical Society, Providence, 2005, 127–173Google Scholar
  3. [3]
    Amberg B., Franciosi S., de Giovanni F., Products of Groups, Oxford Math. Monogr., Oxford University Press, New York, 1992Google Scholar
  4. [4]
    Baaj S., Skandalis G., Vaes S., Measurable Kac cohomology for bicrossed products, Trans. Amer. Math. Soc., 2005, 357(4), 1497–1524MathSciNetMATHCrossRefGoogle Scholar
  5. [5]
    Baumeister B., Factorizations of primitive permutation groups, J. Algebra, 1997, 194(2), 631–653MathSciNetMATHCrossRefGoogle Scholar
  6. [6]
    Caenepeel S., Ion B., Militaru G., Zhu S., The factorization problem and the smash biproduct of algebras and coalgebras, Algebr. Represent. Theory, 2000, 3(1), 19–42MathSciNetMATHCrossRefGoogle Scholar
  7. [7]
    Cap A., Schichl H., Vanžura J., On twisted tensor products of algebras, Comm. Algebra, 1995, 23(12), 4701–4735MathSciNetMATHCrossRefGoogle Scholar
  8. [8]
    Cohn P.M., A remark on the general product of two infinite cyclic groups, Arch. Math. (Basel), 1956, 7(2), 94–99MathSciNetMATHGoogle Scholar
  9. [9]
    Douglas J., On finite groups with two independent generators. I, II, III, IV, Proc. Nat. Acad. Sci. U.S.A., 1951, 37, 604–610, 677–691, 749–760, 808–813MathSciNetMATHCrossRefGoogle Scholar
  10. [10]
    Giudici M., Factorisations of sporadic simple groups, J. Algebra, 2006, 304(1), 311–323MathSciNetMATHCrossRefGoogle Scholar
  11. [11]
    Guccione J.A., Guccione J.J., Valqui C., Twisted planes, Comm. Algebra, 2010, 38(5), 1930–1956MathSciNetMATHCrossRefGoogle Scholar
  12. [12]
    Itô N., Über das Produkt von zwei abelschen Gruppen, Math. Z., 1955, 62, 400–401MathSciNetMATHCrossRefGoogle Scholar
  13. [13]
    Jara Martínez P., López Peña J., Panaite F., Van Oystaeyen F., On iterated twisted tensor products of algebras, Internat. J. Math., 2008, 19(9), 1053–1101MathSciNetMATHCrossRefGoogle Scholar
  14. [14]
    Liebeck M.W., Praeger C.E., Saxl J., The Maximal Factorizations of the Finite Simple Groups and their Automorphism Groups, Mem. Amer. Math. Soc., 86(432), American Mathematical Society, Providence, 1990Google Scholar
  15. [15]
    Liebeck M.W., Praeger C.E., Saxl J., Regular Subgroups of Primitive Permutation Groups, Mem. Amer. Math. Soc., 203 (952), American Mathematical Society, Providence, 2010Google Scholar
  16. [16]
    López Peña J., Navarro G., On the classification and properties of noncommutative duplicates, K-Theory, 2008, 38(2), 223–234MathSciNetMATHCrossRefGoogle Scholar
  17. [17]
    Krötz B., A novel characterization of the Iwasawa decomposition of a simple Lie group, In: Basic Bundle Theory and K-Cohomology Invariants, Lecture Notes in Phys., 726, Springer, Heidelberg, 2007, 195–201Google Scholar
  18. [18]
    Maillet E., Sur les groupes échangeables et les groupes décomposables, Bull. Soc. Math. France, 1900, 28, 7–16MathSciNetMATHGoogle Scholar
  19. [19]
    Masuoka A., Hopf algebra extensions and cohomology, In: New Directions in Hopf Algebras, Math. Sci. Res. Inst. Publ., 43, Cambridge University Press, Cambridge, 2002, 167–209Google Scholar
  20. [20]
    Michor P.W., Knit products of graded Lie algebras and groups, Rend. Circ. Mat. Palermo, 1990, Suppl. 22, 171–175MathSciNetGoogle Scholar
  21. [21]
    Ore O., Structures and group theory. I, Duke Math. J., 1937, 3(2), 149–174MathSciNetCrossRefGoogle Scholar
  22. [22]
    Praeger C.E., Schneider C., Factorisations of characteristically simple groups, J. Algebra, 2002, 255, 198–220MathSciNetMATHCrossRefGoogle Scholar
  23. [23]
    Rédei L., Zur Theorie der faktorisierbaren Gruppen. I, Acta Math. Acad. Sci. Hung., 1950, 1, 74–98MATHCrossRefGoogle Scholar
  24. [24]
    Takeuchi M., Matched pairs of groups and bismash products of Hopf algebras, Comm. Algebra, 1981, 9(8), 841–882MathSciNetMATHCrossRefGoogle Scholar
  25. [25]
    Vaes S., Vainerman L., Extensions of locally compact quantum groups and the bicrossed product construction, Adv. Math., 2003, 175(1), 1–101MathSciNetMATHCrossRefGoogle Scholar
  26. [26]
    Wiegold J., Williamson A.G., The factorisation of the alternating and symmetric groups, Math. Z., 1980, 175(2), 171–179MathSciNetMATHCrossRefGoogle Scholar

Copyright information

© © Versita Warsaw and Springer-Verlag Wien 2011

Authors and Affiliations

  1. 1.Faculty of EngineeringVrije Universiteit BrusselBrusselsBelgium
  2. 2.Faculty of Mathematics and Computer ScienceUniversity of BucharestBucharest 1Romania

Personalised recommendations