Nilpotent Multigroup and its Properties
Keywords:
Multigroup, Commutator subgroups, center of multigroup, Nilpotent, Central seriesAbstract
The theory of multiset generalizes classical set theory which occur as a result of violating basic property of classical set theory that element has only one frequency or can only belong to a set only once. An algebraic structure that generalized crisp group theory over a multiset was established in (Nazmul et al., 2013) and many of its properties have been explored. In this paper, central series of multigroup is a finite chain of normal subgroups computed via commutator submultigroups which aid the study of nilpotent groups and their properties in the multigroup framework. Therefore, it was established that a multigroup is nilpotent if and only if it has a central series generated either via commutator or centre of multigroups. And for every commutative nilpotent multigroup, the nilpotency is one (1). The nilpotency class of the root set is equivalent to the nilpotency of the nilpotent multigroups.
References
Anthony, E. C., Stephen Majewicz & Marcos Zyman (2017) The theory of nilpotent groups
Baer ,R. (1940) Nilpotent groups and their generalizations Trans. Amer. Math. Soc. 47, 393 - 434
Ejegwa, P. A. & Ibrahim A. M., (2020) Some properties of multigroups, Palestine Journal of Mathematics. 9(1), 31 – 47.
Ejegwa, P. A., & Ibrahim A. M., (2017) Normal submultigroups and comultisets of a multigroup, Quasi Related Syst., 25(2), 231 – 244
Ejegwa, P. A., & Ibrahim A. M. (2020) On divisible and pure multigroups and its properties. Open J. Maths. Sci., 4, 377 – 385
Awolola, J. A. (2019) On Cyclic multigroup family. Ratio Mathematica, 37, 61 – 68.
Awolola, J. A. & Ibrahim A. M. (2016) Some results on multigroups. Quasi. Related syst., 24(2) 169 – 177
Michael, B. D., & Ibrahim M. A. (2023) On commutator submultigroups, Kristu Jayanti Journal of Computational Sciences (KJCS), 3(1), 23 – 29. https//doi.org/10.59176/kjcs.v3i1.22.92.
Michael, B. D., Nwagrabe E., & Awolola J.A. (2024) The properties of Commutative multigroup, Faculty of Natural and applied Science Journal of Mathematics and Statistical Computing, 2(1) 39 – 43
Michael, B. D., & Ibrahim M. A. (2019) On the properties of nilpotent group, Nigeria Association of Mathematical Physics, 10, 17 – 19
Nazmul, S. K., Maumba P & Samanta S. K. (2013) On Multisets and multigroups, Annal of Fuzzy Mathematics and informatics, 6(3) 643 – 656.
Singh, D., Ibrahim, A. M., Yohanna T., & Singh J. N. (2007) An overview of the application of multisets, Novi Sad J. Math., 33(2), 73-92.
Syropoulos A. (2001) Mathematics of multisets. Springer Verlag, Berlin, Heidelberg 347 - 358
Adamu, U., & Ibrahim M. A. (2020) Strongly Invariant submultigroups. Theory Appl. Math. Computer Sci. 10(2), 96 – 103.
Zeng L. (2011) Two theorems about nilpotent subgroup. Applied Mathematics, 2, 562 – 564
