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Odd order semidirect extensions of commutative automorphic loops

Přemysl Jedlička (2014)

Commentationes Mathematicae Universitatis Carolinae

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 .

On a class of commutative groupoids determined by their associativity triples

Aleš Drápal (1993)

Commentationes Mathematicae Universitatis Carolinae

Let G = G ( · ) be a commutative groupoid such that { ( a , b , c ) G 3 ; a · b c a b · c } = { ( a , b , c ) G 3 ; a = b c or a b = c } . Then G is determined uniquely up to isomorphism and if it is finite, then card ( G ) = 2 i for an integer i 0 .

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