群的自由积的高可迁表示(英文)  

Highly Transitive Representation of Free Products of Groups

作　　者：孙自行[1] 

机构地区：[1]阜阳师范学院,阜阳236032

出　　处：《南京大学学报（自然科学版）》2005年第4期343-349,共7页Journal of Nanjing University（Natural Science）

基　　金：安徽省自然科学项目(99047217)

摘　　要：可数无限秩的自由格序群是同构于有理数集上的格序置换群A(Q)的2_可迁子群[1,2],THEOREM6.7).McCleary证明了有限秩的自由格序群有一个Q上的2_可迁表示.McCleary给出自由格序群Fη(1<η<0)在Q上有一个o_2_可迁作用[4].这一想法被推广到格序群的自由积.若G是一个L_群,F是基数至少是|F|的无限生成子上的自由群,则自由积G∪H在一个基数|F|的秩域上有一个o_2_可迁表示.Glass和Gurevich则证明了两个可数L_群在Q上有一个o_2_可迁表示[6].证明若G和H是在有理数集Q上有忠实表示的非平凡可数群,则它们的自由积G∪H在Q上有高可迁忠实表示;若G和H是非平凡有限和可数群,且H有一个无限阶元素,则自由积G∪H在自然数集上有高可迁忠实表示.The free lattice-ordered group of a countable infinite rank had been shown to be isomorphic to adoubly transitive subgroup of l-permutationA( Q)of the rational line[1 ,2],theorem6 .7) . McCleary provedthat the free lattice-ordered group of finite ranks possessed a (faithful) doubly transitive representation onQ[3]. McCleary proved that the free lattice-ordered groupFn(1 <η≤0)had been in action of an o-2-transition onQ[4].Theideas were extendedtofree products of l-groups .IfGis anl-group,andFis afreel-group on aninfinite generator set of cardinality at least |F| ,then the free productG∪Hhas a faithful o-2-transitive representation on some ordered fields of cardinality |F|[5]. More recently , Glass and Gurevichshowed that the freel-product of two countablel-groups has had a faithful o-2-transitive representation onQ[6].LetUbe a class of groups ,{Gi|∈I} be afamily ofU.ThenU-free product of thefamily {Gi|i∈I} ,denoted ∪i∈IGiis a groupG∈Utogether with a family of injective homomorphisms {αi∶Gi→G|i∈I}such thati) ∪i∈Iαi(Gi)generatesG;ii) IfH∈Uand{βi∶Gi→H}is a family of homomorphisms ,there exists an unique homomorphismv :G→Hsuch thatβi=vαifor eachi∈I.Our main results areTheorem1 IfGandHare nontrivial countable groups having faithful representations as groups oforder-preserving ofQ,then the productG∪Hhas such a representation which in addition is highly order-transitive .Theorem2 IfGandHare nontrivial finite or countable groups ,and ifHhas an element of infiniteorder ,then the productG∪Hcan be faithfully represented as a highly transitive group onN.We have giventhat ifGandHare nontrivial countable groups havingfaithful representations as groups oforder-preserving permutations ofQ,thentheir free productG∪Hhas such a representation whichin additionis highly order-transitive .IfGandHare nontrivial finite or countable groups and ifHhas an element ofinfinite order ,then free productG∪Hcan be faithfully represented as a highly transitive group on N.

关 键 词：自由积 高可迁 忠实表示 可数群 

分 类 号：O153.1[理学—数学]