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<p>In this article, we study finite coverings of infinite groups by proper normal subgroups of fixed index. We show that free abelian groups, free groups, and fundamental groups of many manifolds and surfaces admit finite coverings by normal subgroups of any fixed index <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(k\ge 2\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>k</mi> <mo>≥</mo> <mn>2</mn> </mrow> </math></EquationSource> </InlineEquation>. We also discuss examples of groups that can be written as a finite union of subgroups of a fixed index but not of all indices.</p>

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Finite coverings of groups via subgroups of fixed index

  • 期刊论文

  • 发表日期：

    2026年7月29日

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  • Deblina Das,
  • Arpan Kabiraj

摘要

In this article, we study finite coverings of infinite groups by proper normal subgroups of fixed index. We show that free abelian groups, free groups, and fundamental groups of many manifolds and surfaces admit finite coverings by normal subgroups of any fixed index \(k\ge 2\) k ≥ 2 . We also discuss examples of groups that can be written as a finite union of subgroups of a fixed index but not of all indices.