Locally compact groups and locally minimal group topologies
Locally compact groups and locally minimal group topologies
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DOI：
10.4064/fm468-3-2018
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发表时间：
2019
影响因子：
0.6
通讯作者：
Wenfei Xi;D. Dikranjan;Wei He;Zhiqiang Xiao
中科院分区：
文献类型：
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作者：
Wenfei Xi;D. Dikranjan;Wei He;Zhiqiang Xiao
。极小群是关于连续同构满足开映射定理的Hausdorff拓扑群G，即每个具有Hausdorff拓扑群H的连续同构G(CID：47)(CID：47)H是拓扑同构。一个拓扑群(G，τ)称为局部极小的，如果存在单位的邻域V，使得对每个Hausdorff群拓扑σ≤τ有V-∈σ，其中一个有σ=τ。极小群和局部紧群都是局部极小的。根据普罗丹诺夫的一个著名定理，finite紧交换群K的每个子群是极小的当且仅当K同构于某个素数p的p进整数群Zp。借助于推广fi定理的结果，我们发现了局部极小与李群和p-进数之间的显著联系：局部紧交换群K的每个子群是局部极小的当且仅当K是一个李群，或者K对某个素数有一个与Zp同构的开子群.在非交换的情况下，我们证明了连通局部紧群的所有子群是局部极小的当且仅当K是李群，解决了[？]中的正问题7.49。
. Minimal groups are the Hausdorff topological groups G satisfying the open mapping theorem with respect to continuous isomorphisms, i.e., every continuous isomorphism G (cid:47) (cid:47) H , with a Hausdorff topological group H , is a topological isomorphism. A topological group ( G, τ ) is called locally minimal if there exists a neighbourhood V of the identity such that for every Hausdorff group topology σ ≤ τ with V ∈ σ one has σ = τ . Minimal groups, as well as locally compact groups, are locally minimal. According to a well known theorem of Prodanov every subgroup of an infinite compact abelian group K is minimal if and only if K is isomorphic to the group Z p of p -adic integers for some prime p . We find a remarkable connection of local minimality to Lie groups and p -adic numbers by means of the following results extending Prodanov’s theorem: every subgroup of a locally compact abelian group K is locally minimal if and only if K is either a Lie group or K has an open subgroup isomorphic to Z p for some prime p . In the nonabelian case we prove that all subgroups of a connected locally compact group are locally minimal if and only if K is a Lie group, resolving in the positive Problem 7.49 from [ ? ].