Central European Journal of Mathematics

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

Schreier type theorems for bicrossed products

  • Ana Agore
  • Gigel Militaru
Research Article


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.


Matched pairs Bicrossed product of groups 


20B05 20B35 20D06 20D40 


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  1. [1]
    Agore A.L., Chirvăsitu A., Ion B., Militaru G., Bicrossed products for finite groups, Algebr. Represent. Theory, 2009, 12(2–5), 481–488MathSciNetzbMATHCrossRefGoogle 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), 1497–1524MathSciNetzbMATHCrossRefGoogle Scholar
  5. [5]
    Baumeister B., Factorizations of primitive permutation groups, J. Algebra, 1997, 194(2), 631–653MathSciNetzbMATHCrossRefGoogle 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), 19–42MathSciNetzbMATHCrossRefGoogle Scholar
  7. [7]
    Cap A., Schichl H., Vanžura J., On twisted tensor products of algebras, Comm. Algebra, 1995, 23(12), 4701–4735MathSciNetzbMATHCrossRefGoogle Scholar
  8. [8]
    Cohn P.M., A remark on the general product of two infinite cyclic groups, Arch. Math. (Basel), 1956, 7(2), 94–99MathSciNetzbMATHGoogle 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, 808–813MathSciNetzbMATHCrossRefGoogle Scholar
  10. [10]
    Giudici M., Factorisations of sporadic simple groups, J. Algebra, 2006, 304(1), 311–323MathSciNetzbMATHCrossRefGoogle Scholar
  11. [11]
    Guccione J.A., Guccione J.J., Valqui C., Twisted planes, Comm. Algebra, 2010, 38(5), 1930–1956MathSciNetzbMATHCrossRefGoogle Scholar
  12. [12]
    Itô N., Über das Produkt von zwei abelschen Gruppen, Math. Z., 1955, 62, 400–401MathSciNetzbMATHCrossRefGoogle 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), 1053–1101MathSciNetzbMATHCrossRefGoogle 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), 223–234MathSciNetzbMATHCrossRefGoogle 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, 7–16MathSciNetzbMATHGoogle 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, 198–220MathSciNetzbMATHCrossRefGoogle Scholar
  23. [23]
    Rédei L., Zur Theorie der faktorisierbaren Gruppen. I, Acta Math. Acad. Sci. Hung., 1950, 1, 74–98zbMATHCrossRefGoogle Scholar
  24. [24]
    Takeuchi M., Matched pairs of groups and bismash products of Hopf algebras, Comm. Algebra, 1981, 9(8), 841–882MathSciNetzbMATHCrossRefGoogle Scholar
  25. [25]
    Vaes S., Vainerman L., Extensions of locally compact quantum groups and the bicrossed product construction, Adv. Math., 2003, 175(1), 1–101MathSciNetzbMATHCrossRefGoogle Scholar
  26. [26]
    Wiegold J., Williamson A.G., The factorisation of the alternating and symmetric groups, Math. Z., 1980, 175(2), 171–179MathSciNetzbMATHCrossRefGoogle 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

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