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Medial magma
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== Bruck–Murdoch–Toyoda theorem == The '''Bruck–Murdoch–Toyoda theorem''' provides the following characterization of medial [[quasigroup]]s. Given an abelian group {{math|''A''}} and two commuting [[group automorphism|automorphisms]] {{math|''φ''}} and {{math|''ψ''}} of {{math|''A''}}, define an operation {{math|•}} on {{math|''A''}} by : {{math|1=''x'' • ''y'' = ''φ''(''x'') + ''ψ''(''y'') + ''c''}}, where {{math|''c''}} some fixed element of {{math|''A''}}. It is not hard to prove that {{math|''A''}} forms a medial quasigroup under this operation. The Bruck–Murdoch-Toyoda theorem states that every medial quasigroup is of this form, i.e. is [[isomorphic]] to a quasigroup defined from an abelian group in this way.{{sfn|ps=|Kuzʹmin|Shestakov|1995}} In particular, every medial quasigroup is [[isotopy of loops|isotopic]] to an abelian group. The result was obtained independently in 1941 by Murdoch and Toyoda.{{sfn|ps=|Murdoch|1941}}{{sfn|ps=|Toyoda|1941}} It was then rediscovered by Bruck in 1944.{{sfn|ps=|Bruck|1944}}
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