Moufang loops of odd order $p_1p_2\dots p_nq^3$ with non-trivial nucleus

Andrew Rajah; Kam-Yoon Chong

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 $M (G, 2)$ .

Possible orders of nonassociative Moufang loops

Orin Chein, Andrew Rajah (2000)

Commentationes Mathematicae Universitatis Carolinae

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The paper surveys the known results concerning the question: “For what values of $n$ does there exist a nonassociative Moufang loop of order $n$ ?” Proofs of the newest results for $n$ odd, and a complete resolution of the case $n$ even are also presented.

Displaying similar documents to “Moufang loops of odd order p 1 p 2 ⋯ p n q 3 with non-trivial nucleus”

On the uniqueness of loops M ( G , 2 )

Possible orders of nonassociative Moufang loops

Displaying similar documents to “Moufang loops of odd order $p_{1} p_{2} \dots p_{n} q^{3}$ with non-trivial nucleus”

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