A class of Bol loops with a subgroup of index two

Petr Vojtěchovský

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 $(x, y) \mapsto {(x^{i} y^{j})}^{k}$ , where $i, j, k \in {- 1, 1}$ . Define a new multiplication on $G \times C_{2}$ by assigning one of the above 8 multiplications to each quarter $(G \times {i}) \times (G \times {j})$ , for $i, j \in C_{2}$ . 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 $(G, \cdot)$ is a group, and the operation $(*)$ is defined by $x * y = x \cdot y^{- 1}$ 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 $(G, \cdot)$ can be constructed, so that in effect $(G, \cdot)$ 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...

On multiplication groups of left conjugacy closed loops

Aleš Drápal (2004)

Commentationes Mathematicae Universitatis Carolinae

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A loop $Q$ is said to be left conjugacy closed (LCC) if the set ${L_{x}; x \in Q}$ is closed under conjugation. Let $Q$ be such a loop, let $ℒ$ and $ℛ$ be the left and right multiplication groups of $Q$ , respectively, and let $Inn Q$ be its inner mapping group. Then there exists a homomorphism $ℒ \to Inn Q$ determined by $L_{x} \mapsto R_{x}^{- 1} L_{x}$ , and the orbits of $[ℒ, ℛ]$ coincide with the cosets of $A (Q)$ , the associator subloop of $Q$ . All LCC loops of prime order are abelian groups.

Bol-loops of order $3 \cdot 2^{n}$

Daniel Wagner, Stefan Wopperer (2007)

Acta Universitatis Palackianae Olomucensis. Facultas Rerum Naturalium. Mathematica

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In this article we construct proper Bol-loops of order $3 \cdot 2^{n}$ using a generalisation of the semidirect product of groups defined by Birkenmeier and Xiao. Moreover we classify the obtained loops up to isomorphism.

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

On the uniqueness of loops M ( G , 2 )

Right division in Moufang loops

On multiplication groups of left conjugacy closed loops

Bol-loops of order 3 · 2 n

On the uniqueness of loops $M (G, 2)$

Bol-loops of order $3 \cdot 2^{n}$