Algebraically determined topologies on permutation groups
Algebraically determined topologies on permutation groups
复制标题
置换群的代数确定拓扑
DOI：
10.1016/j.topol.2012.04.010
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发表时间：
2011
期刊：
影响因子：
--
通讯作者：
I. Protasov
中科院分区：
文献类型：
--
作者：
T. Banakh;Igor Guran;I. Protasov
本文回答了Dikran Dikranjan关于集合X的双射群S（X）（和Sω（X））上的代数决定拓扑的几个问题。特别地，我们证实了Dikranjan的猜想，证明了S（X）的每个子群G <$Sω（X）上的点态收敛拓扑Tp是G上的粗Hausdorff群拓扑（更一般地说，是使G成为[半]-拓扑群的粗T1-拓扑），且Tp与G上的Zerkki拓扑和Markov拓扑Z G和MG一致.我们讨论了Dikranjan的另一个问题，证明了对称群G=S（X）上的中心化拓扑TG是离散的当且仅当|X| 200c.另一方面，我们证明了对于S（X）的子群G <$Sω（X），中心化拓扑TG与拓扑Tp=MG= ZG重合当且仅当G=Sω（X）.我们还证明了群Sω（X）在每个Hausdorff移位不变拓扑中是σ-离散的。
In this paper we answer several questions of Dikran Dikranjan about algebraically determined topologies on the groups S(X) (and Sω(X)) of (finitely supported) bijections of a set X. In particular, confirming conjecture of Dikranjan, we prove that the topology Tpof pointwise convergence on each subgroup G⊃Sω(X) of S(X) is the coarsest Hausdorff group topology on G (more generally, the coarsest T1-topology which turns G into a [semi]-topological group), and Tpcoincides with the Zariski and Markov topologies ZGand MGon G. Answering another question of Dikranjan, we prove that the centralizer topology TGon the symmetric group G=S(X) is discrete if and only if |X|⩽c. On the other hand, we prove that for a subgroup G⊃Sω(X) of S(X) the centralizer topology TGcoincides with the topologies Tp=MG=ZGif and only if G=Sω(X). We also prove that the group Sω(X) is σ-discrete in each Hausdorff shift-invariant topology.
影响因子：
0.5
作者：
Dikranjan, Dikran;Shakhmatov, Dmitri
通讯作者：
Shakhmatov, Dmitri