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Comptes Rendus. Mathématique

Algèbre
Group extensions and marginal series of pair of groups
Comptes Rendus. Mathématique, Tome 359 (2021) no. 5, pp. 631-638.

In this article, using the concept of generalized Baer-invariant of a pair of groups, we establish some related isomorphisms between lower marginal quotient pairs of groups, which are generalized versions of some isomorphisms of Stallings. We also derive a result for the pair (𝒱.𝒲,𝒳) to be an ultra Hall pair for special varieties of groups. This result generalizes that of Fung in 1977, which has roots in Philip Hall’s criterion on nilpotency.

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DOI : https://doi.org/10.5802/crmath.212
Classification : 20E10,  20F19,  20J05
@article{CRMATH_2021__359_5_631_0,
     author = {Mohammad Reza Rismanchian},
     title = {Group extensions and marginal series of pair of groups},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {631--638},
     publisher = {Acad\'emie des sciences, Paris},
     volume = {359},
     number = {5},
     year = {2021},
     doi = {10.5802/crmath.212},
     language = {en},
}
Mohammad Reza Rismanchian. Group extensions and marginal series of pair of groups. Comptes Rendus. Mathématique, Tome 359 (2021) no. 5, pp. 631-638. doi : 10.5802/crmath.212. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.212/

[1] Graham Ellis The schur multiplier of a pair of groups, Appl. Categ. Struct., Volume 6 (1998) no. 3, pp. 355-371 | Article | MR 1641859 | Zbl 0948.20026

[2] W. K. H. Fung Some theorems of Hall type, Arch. Math., Volume 27 (1977), pp. 9-20 | Article | MR 435218 | Zbl 0363.20023

[3] Philip Hall The classification of prime power groups, J. Reine Angew. Math., Volume 182 (1940), pp. 130-141 | MR 3389

[4] N. S. Hekster Varities of groups and isologisms, J. Aust. Math. Soc., Volume 46 (1989), pp. 22-60 | Article | MR 966283 | Zbl 0667.20021

[5] Peter J. Hilton; Urs Stammbach A Course in Homological Algebra, Graduate Texts in Mathematics, 4, Springer, 1970 | Zbl 0238.18006

[6] J. A. Hulse; John C. Lennox Marginal series in groups, Proc. R. Soc. Edinb., Sect. A, Math., Volume 76 (1977), pp. 139-154 | Article | Zbl 0364.20038

[7] C. R. Leedham-Green; Susan McKay Baer-invariant, isologism, varietal laws and homology, Acta Math., Volume 137 (1976), pp. 99-150 | Article | MR 435250 | Zbl 0364.20036

[8] Hanna Neumann Varieties of Groups, Ergebnisse der Mathematik und ihrer Grenzgebiete, 37, Springer, 1967 | MR 215899 | Zbl 0251.20001

[9] Mohammad Reza Rismanchian 𝒱-nilpotent groups and 5-term exact sequence, Commun. Algebra, Volume 42 (2014) no. 4, pp. 1559-1564 | Article | MR 3169650 | Zbl 1291.20026

[10] Mohammad Reza Rismanchian; Mehdi Araskhan Some inequalities for the dimension of the Schur multiplier of a pair of (nilpotent) Lie Algebras, J. Algebra, Volume 352 (2012) no. 1, pp. 173-179 | Article | MR 2862180 | Zbl 1261.17010

[11] Mohammad Reza Rismanchian; Mehdi Araskhan Some properties of the c-nilpotent multiplier and c-covers of Lie algebras, Algebra Colloq., Volume 21 (2014) no. 3, pp. 421-426 | Article | MR 3224666 | Zbl 1330.17016

[12] Derek J. S. Robinson A Course in the Theory of Groups, Graduate Texts in Mathematics, 80, Springer, 1995 | Zbl 0836.20001

[13] Issai Schur Über die Darstellung der endlichen Gruppen durch gebrochene lineare Substitutionen, J. für Math., Volume 127 (1904), pp. 20-50 | Zbl 35.0155.01

[14] Issai Schur Untersuchungen Über die Darstellung der endlichen Gruppen durch gebrochene lineare Substitutionen, J. Reine Angew. Math., Volume 132 (1907), pp. 85-137 | Zbl 38.0174.02

[15] John Stallings Homology and central series of groups, J. Algebra, Volume 2 (1965), pp. 170-181 | Article | MR 175956