A group G is said to be n-abelian, if (xy)n=xnyn, for any x,y ∈in G and a positive integer n. In 1979, Fay and Waals introduced the n-potent and the n-center subgroups of a group G, denoted by Gn and Zn(G), respectively. In this paper, we show that the index of the n-center is bounded by an order power of the n-potent subgroup, for some classes of groups. In fact for all n-abelian groups G with finite n-potent subgroup, we prove that if G/Zn(G) is finitely generated, then [G : Zn(G)] ≤ |Gn|d(G/Zn(G)). Moreover, we conclude that [G : Zn(G)] ≤ |Gn|2log2|Gn|, for some n-capable group G.
Pourmirzaei, A. (2021). On the order of the n-center factor subgroup of an n-abelian group. Analytical and Numerical Solutions for Nonlinear Equations, 6(2), 303-308. doi: 10.22128/gadm.2021.486.1061
MLA
Azam Pourmirzaei. "On the order of the n-center factor subgroup of an n-abelian group", Analytical and Numerical Solutions for Nonlinear Equations, 6, 2, 2021, 303-308. doi: 10.22128/gadm.2021.486.1061
HARVARD
Pourmirzaei, A. (2021). 'On the order of the n-center factor subgroup of an n-abelian group', Analytical and Numerical Solutions for Nonlinear Equations, 6(2), pp. 303-308. doi: 10.22128/gadm.2021.486.1061
VANCOUVER
Pourmirzaei, A. On the order of the n-center factor subgroup of an n-abelian group. Analytical and Numerical Solutions for Nonlinear Equations, 2021; 6(2): 303-308. doi: 10.22128/gadm.2021.486.1061