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• W2964275744 abstract "An abelian group <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=upper A> <mml:semantics> <mml:mi>A</mml:mi> <mml:annotation encoding=application/x-tex>A</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is said to be cancellable if whenever <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=upper A circled-plus upper G> <mml:semantics> <mml:mrow> <mml:mi>A</mml:mi> <mml:mo>⊕<!-- ⊕ --></mml:mo> <mml:mi>G</mml:mi> </mml:mrow> <mml:annotation encoding=application/x-tex>A oplus G</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is isomorphic to <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=upper A circled-plus upper H> <mml:semantics> <mml:mrow> <mml:mi>A</mml:mi> <mml:mo>⊕<!-- ⊕ --></mml:mo> <mml:mi>H</mml:mi> </mml:mrow> <mml:annotation encoding=application/x-tex>A oplus H</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=upper G> <mml:semantics> <mml:mi>G</mml:mi> <mml:annotation encoding=application/x-tex>G</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is isomorphic to <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=upper H> <mml:semantics> <mml:mi>H</mml:mi> <mml:annotation encoding=application/x-tex>H</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. We show that the index set of cancellable rank 1 torsion-free abelian groups is <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=normal upper Pi 4 Superscript 0> <mml:semantics> <mml:msubsup> <mml:mi mathvariant=normal>Π<!-- Π --></mml:mi> <mml:mn>4</mml:mn> <mml:mn>0</mml:mn> </mml:msubsup> <mml:annotation encoding=application/x-tex>Pi ^0_4</mml:annotation> </mml:semantics> </mml:math> </inline-formula> <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=m> <mml:semantics> <mml:mi>m</mml:mi> <mml:annotation encoding=application/x-tex>m</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-complete, showing that the classification by Fuchs and Loonstra cannot be simplified. For arbitrary non-finitely generated groups, we show that the cancellation property is <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=normal upper Pi 1 Superscript 1> <mml:semantics> <mml:msubsup> <mml:mi mathvariant=normal>Π<!-- Π --></mml:mi> <mml:mn>1</mml:mn> <mml:mn>1</mml:mn> </mml:msubsup> <mml:annotation encoding=application/x-tex>Pi ^1_1</mml:annotation> </mml:semantics> </mml:math> </inline-formula> <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=m> <mml:semantics> <mml:mi>m</mml:mi> <mml:annotation encoding=application/x-tex>m</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-hard; we know of no upper bound, but we conjecture that it is <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=normal upper Pi 2 Superscript 1> <mml:semantics> <mml:msubsup> <mml:mi mathvariant=normal>Π<!-- Π --></mml:mi> <mml:mn>2</mml:mn> <mml:mn>1</mml:mn> </mml:msubsup> <mml:annotation encoding=application/x-tex>Pi ^1_2</mml:annotation> </mml:semantics> </mml:math> </inline-formula> <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=m> <mml:semantics> <mml:mi>m</mml:mi> <mml:annotation encoding=application/x-tex>m</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-complete." @default.
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• W2964275744 date "2019-05-01" @default.
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• W2964275744 title "Characterizations of cancellable groups" @default.
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