自由亚交换群的直积的检验元素

Test Elements of Direct Products of Free Metabelian Groups

设Sri(i=1,2,…,n)为秩ri的自由亚交换群,G=Sr1×Sr2…×Srn为自由亚交换群的直积,本文证明了G有检验元素的充分必要条件为ri=2(i=1,2,…,n)．同时,还证明了g=(g1,g2,…,gn)为G的检验元素的充分必要条件是:gi∈S′2-1(i= 1,2,…,n),且{g1,g2,…,gn}为独立集．此外,我们给出了一类具体的检验元素．

Let Sri, （i = 1,2,... ,n） be free metabelian groups of rank ri, G = Sr1 × Sr2 × ... × Srn be direct product of free metabelian groups. In this paper, we prove that G has test element if and only if ri = 2 （i = 1,2,... ,n）. Meanwhile, we prove that g = （g1, g2,..., gn） is a test element of G if and only if gi ∈ S2^′ - 1 （i = 1, 2,..., n）, and {g1,g2,..,gn} is an independent set.