An Upper Bound for the Index of the Second n-Center Subgroup of An n-Abelian Group

Document Type : Research Paper

Author

Hakim Sabzevari University

Abstract

Let n be a positive integer. A group G is said to be n-abelian, if (xy)n = xnyn, for any x, y ∈ G. In 1979, Fay and Waals introduced the n- potent and the n-center subgroups of a group G, as Gn = ⟨[x, yn]|x, y ∈ G⟩, Zn(G) = {x ∈ G|xyn = ynx, ∀y ∈ G}, respectively. Also, the second n-center subgroup, Zn ∈ (G), is defined by Zn 2 (G)/Zn(G) = Zn(G/Zn(G)). In this paper, we give an upper bound for the index of the second n-center subgroup of any n-abelian group G in terms of the order of n-potent subgroup Gn. 

Keywords

References

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Analytical and Numerical Solutions for Nonlinear Equations
Volume 6, Issue 2
November 2021
Pages 263-368

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History

  • Receive Date: 25 December 2021
  • Revise Date: 27 December 2021
  • Accept Date: 15 January 2022

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