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Displaying similar documents to “A class of Bol loops with a subgroup of index two”

On the uniqueness of loops M ( G , 2 )

Petr Vojtěchovský (2003)

Commentationes Mathematicae Universitatis Carolinae

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Let G be a finite group and C 2 the cyclic group of order 2. Consider the 8 multiplicative operations ( x , y ) ( x i y j ) k , where i , j , k { - 1 , 1 } . Define a new multiplication on G × C 2 by assigning one of the above 8 multiplications to each quarter ( G × { i } ) × ( G × { j } ) , for i , j C 2 . If the resulting quasigroup is a Bol loop, it is Moufang. When G is nonabelian then exactly four assignments yield Moufang loops that are not associative; all (anti)isomorphic, known as loops M ( G , 2 ) .

Right division in Moufang loops

Maria de Lourdes M. Giuliani, Kenneth Walter Johnson (2010)

Commentationes Mathematicae Universitatis Carolinae

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If ( G , · ) is a group, and the operation ( * ) is defined by x * y = x · y - 1 then by direct verification ( G , * ) is a quasigroup which satisfies the identity ( x * y ) * ( z * y ) = x * z . Conversely, if one starts with a quasigroup satisfying the latter identity the group ( G , · ) can be constructed, so that in effect ( G , · ) is determined by its right division operation. Here the analogous situation is examined for a Moufang loop. Subtleties arise which are not present in the group case since there is a choice of defining identities and the identities...