Article in volume
Authors:
Noureddine Snanou
Department of Mathematics
Sciences Faculty, Mohammed First University
Oujda, Morocco.
email: n.snanou@ump.ac.ma
Title:
On the isomorphism problem for knit products
PDFSource:
Discussiones Mathematicae - General Algebra and Applications 45(1) (2025) 177-189
Received: 2023-12-21 , Revised: 2024-05-19 , Accepted: 2024-05-20 , Available online: 2024-08-22 , https://doi.org/10.7151/dmgaa.1460
Abstract:
In this paper, we classify up to isomorphism the groups that can be represented as knit products of two groups. More precisely, some necessary and sufficient conditions for two knit products to be isomorphic are given. We mainly deal with isomorphisms leaving one of the two factors or even both invariant. In particular, we decide under some conditions how the knit products arise as split extensions. Furthermore, the decomposition of unfaithful knit products is investigated.
Keywords:
knit product, factorization problem, lower isomorphic, upper isomorphic, diagonally isomorphic
References:
- A.L. Agore, A. Chirv$\breve{a}$situ, B. Ion and G. Militaru, Bicrossed products for finite groups, Algebr. Represent. Theory 12 (2009) 481–488.
https://doi.org/10.1007/s10468-009-9145-6 - F. Ateş and A.S. Çevik, Knit products of some groups and their applications, Rend. Sem. Mat. Univ. Padova 121 (2009) 1–11.
https://doi.org/10.4171/RSMUP/121-1 - M.G. Brin, On the Zappa-Szép product, Communications in Algebra 33(2) (2005) 393–424.
https://doi.org/10.1081/AGB-200047404 - J. Douglas, On finite groups with two independent generators. I, II, III, IV, Proc. Nat. Acad. Sci. U.S.A. 37 (1951) 604–610, 677–691, 749–760, 808–813, 10.1073/pnas.37.10.677.
https://doi.org/10.1073/pnas.37.9.604 - S. Eilenberg and S. MacLane, Group extensions and homology, Ann. Math., Second Series 43(4) (1942) 757–831.
https://doi.org/10.2307/1968966 - P. Hall, A characteristic property of soluble groups, J. London Math. Soc. 12 (1937) 198–200.
https://doi.org/10.1112/jlms/s1-12.2.198 - B. Huppert, Über das Produkt von paarweise vertauschbaren zyklischen Gruppen, Math. Z. 58 (1953) 243–264.
https://doi.org/10.1007/BF01174144 - S. MacLane, Homology (Springer-Verlag, Berlin, Göttingen, Heidelberg, 1963).
- L. Rédei, Zur Theorie der faktorisierbaren Gruppen I, Acta Math. Acad. Sci. Hungar. 1 (1950) 74–98.
https://doi.org/10.1007/BF02022554 - O. Schreier, Über die Erweiterung von Gruppen I, Monatsh. Math. Phys. 34 (1926) 165–180.
https://doi.org/10.1007/BF01694897 - N. Snanou and M.E. Charkani, On the isomorphism problem for split extensions, J. Algebra Appl. 23(2) (2024) 2430002.
https://doi.org/10.1142/S0219498824300022 - N. Snanou, On non-split abelian extensions II, Asian-Eur. J. Math. 14(9) (2021) 2150164.
https://doi.org/10.1142/S1793557121501643 - N. Snanou, On the isomorphism problem for central extensions I, Proc. Jangjeon Math. Soc. 27(2) (2024) 101–109.
https://doi.org/10.17777/pjms2024.27.2.101 - N. Snanou, On the Isomorphism Problem for Central Extensions II, Eur. J. Pure Appl. Math. 17(2) (2024) 956–968.
https://doi.org/10.29020/nybg.ejpam.v17i2.5118 - J. Szép, Über die als Produkt zweier Untergruppen darstellbaren endlichen Gruppen, Comment. Math. Helv. 22 (1949) 31–33.
https://doi.org/10.1007/BF02568046 - J. Szép and L. Rédei, On factorisable groups, Acta Univ. Szeged. Sect. Sci. Math. 13 (1950) 235–238.
- J. Szép, Zur Theorie der endlichen einfachen Gruppen, Acta Sci. Math. Szeged 14 (1951) 111–112.
- K.R. Yacoub, On general products of two finite cyclic groups one of which being of order $p^{2}$, Publ. Math. Debrecen 6 (1959) 26–39.
- G. Zappa, Sulla costruzione dei gruppi prodotto di due dati sottogruppi permutabili tra loro, in: Atti Secondo Congresso Un. Mat. Ital. Bologna, 1940, (Edizioni Cremonense, Rome 1942) 119–125.
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