The normalizer conjecture in the alternative case (Q2773015)
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scientific article; zbMATH DE number 1709134
| Language | Label | Description | Also known as |
|---|---|---|---|
| English | The normalizer conjecture in the alternative case |
scientific article; zbMATH DE number 1709134 |
Statements
17 September 2002
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unit loops
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units
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torsion loops
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integral loop rings
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Moufang loops
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normalizers
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centers
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Hamiltonian 2-loops
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The normalizer conjecture in the alternative case (English)
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Let \(L\) be a torsion loop for which the integral loop ring \(\mathbb{Z} L\) is an alternative, not associative ring. The invertible elements (units) in \(\mathbb{Z} L\) form a Moufang loop, denoted by \(U(\mathbb{Z} L)\). Let \(N_U(L)\) denote the normalizer of \(L\) in \(U(\mathbb{Z} L)\) and \(Z(U)\) the center of \(U(\mathbb{Z} L)\). The authors prove that \(N_U(L)=Z(U)L\) and that \(U(\mathbb{Z} L)\) has central height 1, unless \(L\) is a Hamiltonian 2-loop.
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