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: G → H 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 G ≅H α′⋈β′ (G, *). Moreover, if we fix the group H and the automorphism σ ∈ Aut H then any σ-invariant isomorphism H α⋈β G ≅ H 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.
Similar content being viewed by others
References
Agore A.L., Chirvăsitu A., Ion B., Militaru G., Bicrossed products for finite groups, Algebr. Represent. Theory, 2009, 12(2–5), 481–488
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, 127–173
Amberg B., Franciosi S., de Giovanni F., Products of Groups, Oxford Math. Monogr., Oxford University Press, New York, 1992
Baaj S., Skandalis G., Vaes S., Measurable Kac cohomology for bicrossed products, Trans. Amer. Math. Soc., 2005, 357(4), 1497–1524
Baumeister B., Factorizations of primitive permutation groups, J. Algebra, 1997, 194(2), 631–653
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), 19–42
Cap A., Schichl H., Vanžura J., On twisted tensor products of algebras, Comm. Algebra, 1995, 23(12), 4701–4735
Cohn P.M., A remark on the general product of two infinite cyclic groups, Arch. Math. (Basel), 1956, 7(2), 94–99
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, 808–813
Giudici M., Factorisations of sporadic simple groups, J. Algebra, 2006, 304(1), 311–323
Guccione J.A., Guccione J.J., Valqui C., Twisted planes, Comm. Algebra, 2010, 38(5), 1930–1956
Itô N., Über das Produkt von zwei abelschen Gruppen, Math. Z., 1955, 62, 400–401
Jara Martínez P., López Peña J., Panaite F., Van Oystaeyen F., On iterated twisted tensor products of algebras, Internat. J. Math., 2008, 19(9), 1053–1101
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, 1990
Liebeck M.W., Praeger C.E., Saxl J., Regular Subgroups of Primitive Permutation Groups, Mem. Amer. Math. Soc., 203 (952), American Mathematical Society, Providence, 2010
López Peña J., Navarro G., On the classification and properties of noncommutative duplicates, K-Theory, 2008, 38(2), 223–234
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, 195–201
Maillet E., Sur les groupes échangeables et les groupes décomposables, Bull. Soc. Math. France, 1900, 28, 7–16
Masuoka A., Hopf algebra extensions and cohomology, In: New Directions in Hopf Algebras, Math. Sci. Res. Inst. Publ., 43, Cambridge University Press, Cambridge, 2002, 167–209
Michor P.W., Knit products of graded Lie algebras and groups, Rend. Circ. Mat. Palermo, 1990, Suppl. 22, 171–175
Ore O., Structures and group theory. I, Duke Math. J., 1937, 3(2), 149–174
Praeger C.E., Schneider C., Factorisations of characteristically simple groups, J. Algebra, 2002, 255, 198–220
Rédei L., Zur Theorie der faktorisierbaren Gruppen. I, Acta Math. Acad. Sci. Hung., 1950, 1, 74–98
Takeuchi M., Matched pairs of groups and bismash products of Hopf algebras, Comm. Algebra, 1981, 9(8), 841–882
Vaes S., Vainerman L., Extensions of locally compact quantum groups and the bicrossed product construction, Adv. Math., 2003, 175(1), 1–101
Wiegold J., Williamson A.G., The factorisation of the alternating and symmetric groups, Math. Z., 1980, 175(2), 171–179
Author information
Authors and Affiliations
Corresponding author
About this article
Cite this article
Agore, A., Militaru, G. Schreier type theorems for bicrossed products. centr.eur.j.math. 10, 722–739 (2012). https://doi.org/10.2478/s11533-011-0128-6
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.2478/s11533-011-0128-6