# Some remarks on goursat lemma

Document Type : Research Paper

Author

Department of mathematics, University of Yaounde 1, Yaounde, Cameroon.

10.29252/as.2021.2022

Abstract

In this article,we give a characterization of containment of subgroups in a direct product $A\times B\times C$. Other potential generalizations are investigated and applications characterizing different types of groups and modules are given. Most of applications are simple while somewhat deeper applications occur in the case of cyclic modules.

Keywords

#### References

[1] B. R. Amougou Mbarga, Triangular Scheme Revisited in the Light of n-permutable Categories, Earthline Journal of Mathematical Sciences ISSN(Online), 6 No. 1 (2021) 105-116.
[2] B. R. Amougou Mbarga, Anticommutativity and n-schemes, Earthline Journal of Mathematical Sciences ISSN (Online), 6 No. 1 (2021).
[3] D. D. Anderson and V. Camillo, Subgroups of direct products of groups, ideals and subrings of direct products of rings, and Goursat's lemma, Rings, modules and representations, 480 (2009) 1-12.
[4] R. Baer, Der Kern eine charakteristiche Untergruppe, Compos. Math., 1 (1934) 254-283.
[5] A. Carboni, J. Lambek and M. C. Pedicchio, Diagram chasing in Mal'cev categories, Appl. Algebra, 69 (1990) 271-284.
[6] J. Evan, Permutability of subgroups of G ×H that are direct products of subgroups of the direct factors, Archiv. Math. (Basel), 77 No. 6 (2001) 449-455.
[7] J. Evan, Permutable Diagonal-type Subgroups of G ×H, Glasg. Math. J., 45 No. 1 (2003) 73-77.
[8] J. F. Farriel and S. -Lack, For which categories does one have a Goursat lemma?, 2010.
[9] E. Goursat, Sur les substitutions orthogonales et les divisions réguliéres de l'espace, Ann. Sci. l'École Norm. Sup., 6 (1889) 9-102.
[10] J. Lambek, Goursat's theorem and the Zassenhaus lemma, Canad. J. Math., 10 (1958) 45-56.
[11] J. Lambek, On the ubiquity of Mal'cev operarations, Contemp. Math., 131 (1993) 135-135.
[12] S. Lang and T. E. Algebra, Addition-Wesley, MR0197234 (33: 5416), 1993.
[13] D. C. Lewis, Containment of Subgroups in a Direct Product of Groups, Doctoral dissertation, State University of New York at Binghamton, Department of Mathematical Sciences 2011.
[14] O. Oluwafunmilayo and M. EniOluwafe, On counting subgroups for a class of finite nonabelian p-groups and related problems, IMHOTEP: Afr. J. Pure Appl. Math., 4 No. 1 (2017) 34-43.
[15] D. Sen, K. Bauer and P. Zvengrowski, A generalized Goursat lemma, Tatra Mt. Math. publ., 64 (2015) 1-19.
[16] J. J. OĆonnor and E. F. Roberston, Edourd Jean Baptiste Goursat, MacTutor, History of Mathematics, http:// www-history.mcs.st-andrews.ac.uk/Biographiies/Goursat.htm, August 2006.
[17] J. J. Rotman, An introduction to the theory of groups, (forth edition), in: Grad. Texts in Math., 148, Springer-Verlag, New York, 1995.
[18] R. Schmidt, Subgroup lattices of groups, (de Gruyter, Berlin, 1994).
[19] L. TÒTH, Subgroups of finite abelian groups having rank two via goursat's lemma, Tatra Mt. Math. Publ., 59 (2014) 93-103.
[20] M. Tǎrnǎuceanu, Counting subgroups for a class of fnite nonabelian p-groups, Analele Universitaǎatii de Vest; Timisoara Seria Mathematicǎ-InformaticǎXLVI, 1 (2008) 147-152.
[21] V. M. Usenko, Subgroups of semidirect products, Ukrain. Mat. Zh., 43 No. 7 (1991) 982-988.