BOUNDING THE ORDER OF THE NILPOTENT RESIDUAL OF A FINITE GROUP

作者：de Andrade Agenor Freitas; Shumyatsky Pavel^*

来源：Bulletin of the Australian Mathematical Society, 2016, 94(2): 273-277.

DOI：10.1017/S0004972716000150

• finite groups
• nilpotent residual

The last term of the lower central series of a finite group G is called the nilpotent residual. It is usually denoted by gamma(infinity)(G). The lower Fitting series of G is defined by D-0(G) = G and Di+1(G) = gamma(infinity)(Di(G)) for i = 0, 1, 2..... These subgroups are generated by so-called coprime commutators gamma(k)* and delta(k)* in elementsof G. More precisely, the set of coprime commutators gamma(k)* generates gamma(infinity) (G) whenever k >= 2 while the set delta(k)* generates D-k(G) for k >= 0. The main result of this article is the following theorem: let m be a positive integer and G a finite group. Let X subset of G be either the set of all gamma(k)*-commutators for some fixed k >= 2 or the set of all delta(k)*-commutators for some fixed k >= 1. Suppose that the size of a(X) is at most m for any a 2 G. Then the order of < X > is (k, m)-bounded.

• 出版日期2016-10