Dihedral-like constructions of automorphic loops

Mouna Aboras

Displaying similar documents to “Dihedral-like constructions of automorphic loops”

On centerless commutative automorphic loops

Gábor P. Nagy (2014)

Commentationes Mathematicae Universitatis Carolinae

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In this short paper, we survey the results on commutative automorphic loops and give a new construction method. Using this method, we present new classes of commutative automorphic loops of exponent $2$ with trivial center.

Odd order semidirect extensions of commutative automorphic loops

Přemysl Jedlička (2014)

Commentationes Mathematicae Universitatis Carolinae

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We analyze semidirect extensions of middle nuclei of commutative automorphic loops. We find a less complicated conditions for the semidirect construction when the middle nucleus is an odd order abelian group. We then use the description to study extensions of orders $3$ and $5$ .

A class of Bol loops with a subgroup of index two

Petr Vojtěchovský (2004)

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 $i, j \in C_{2}$ . We describe all situations in which the resulting quasigroup is a Bol loop. This paper also corrects an error in P. Vojtěchovsk’y: On the uniqueness of loops $M (G, 2)$ .

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 $M (G, 2)$ .

Displaying similar documents to “Dihedral-like constructions of automorphic loops”

On centerless commutative automorphic loops

Odd order semidirect extensions of commutative automorphic loops

A class of Bol loops with a subgroup of index two

On the uniqueness of loops M ( G , 2 )

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