On lattices of closed subgroups in the group of infinite triangular matrices over a field

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• 作者：Agnieszka Bier ^{agnieszka.bier@polsl.pl" class="auth_mail" title="E-mail the corresponding author}
• 关键词：20E15 ; 20F12 ; 20F14 ; 20G15
• 刊名：Linear Algebra and its Applications
• 出版年：2015
• 出版时间：15 November 2015
• 年：2015
• 卷：485
• 期：Complete
• 页码：132-152
• 全文大小：485 K

文摘

We investigate a special type of closed subgroups of the topological group class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0024379515004462&_mathId=si1.gif&_user=111111111&_pii=S0024379515004462&_rdoc=1&_issn=00243795&md5=91774733fd17c6d86eda6d3ee573d2ad" title="Click to view the MathML source">UT(∞,K)class="mathContainer hidden">class="mathCode">$\mathrm{UT}(\infty ,K)$ of infinite-dimensional unitriangular matrices over a field K   (class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0024379515004462&_mathId=si2.gif&_user=111111111&_pii=S0024379515004462&_rdoc=1&_issn=00243795&md5=8ba2a4247f69b48c5ac2efa3b9708e80" title="Click to view the MathML source">|K|>2class="mathContainer hidden">class="mathCode">$|K|>2$), considered with the natural inverse limit topology. Namely, we generalize the concept of partition subgroups introduced in [23] and define partition subgroups in class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0024379515004462&_mathId=si1.gif&_user=111111111&_pii=S0024379515004462&_rdoc=1&_issn=00243795&md5=91774733fd17c6d86eda6d3ee573d2ad" title="Click to view the MathML source">UT(∞,K)class="mathContainer hidden">class="mathCode">$\mathrm{UT}(\infty ,K)$. We show that they are all closed and discuss the problem of their invariance to various group homomorphisms. We prove that a characteristic subgroup of class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0024379515004462&_mathId=si1.gif&_user=111111111&_pii=S0024379515004462&_rdoc=1&_issn=00243795&md5=91774733fd17c6d86eda6d3ee573d2ad" title="Click to view the MathML source">UT(∞,K)class="mathContainer hidden">class="mathCode">$\mathrm{UT}(\infty ,K)$ is necessarily a partition subgroup and characterize the lattices of characteristic and fully characteristic subgroups in class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0024379515004462&_mathId=si1.gif&_user=111111111&_pii=S0024379515004462&_rdoc=1&_issn=00243795&md5=91774733fd17c6d86eda6d3ee573d2ad" title="Click to view the MathML source">UT(∞,K)class="mathContainer hidden">class="mathCode">$\mathrm{UT}(\infty ,K)$. We conclude with some implications of the given characterization on verbal structure of class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0024379515004462&_mathId=si1.gif&_user=111111111&_pii=S0024379515004462&_rdoc=1&_issn=00243795&md5=91774733fd17c6d86eda6d3ee573d2ad" title="Click to view the MathML source">UT(∞,K)class="mathContainer hidden">class="mathCode">$\mathrm{UT}(\infty ,K)$ and class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0024379515004462&_mathId=si30.gif&_user=111111111&_pii=S0024379515004462&_rdoc=1&_issn=00243795&md5=be122fa7a217866726bcdedf850857f1" title="Click to view the MathML source">T(∞,K)class="mathContainer hidden">class="mathCode">$\mathrm{T}(\infty ,K)$ and use some topological properties to discuss the problem of the width of verbal subgroups in groups defined over a finite field K.