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Displaying similar documents to “Clifford algebras, Möbius transformations, Vahlen matrices, and B -loops”

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

Pseudoautomorphisms of Bruck loops and their generalizations

Mark Greer, Michael Kinyon (2012)

Commentationes Mathematicae Universitatis Carolinae

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We show that in a weak commutative inverse property loop, such as a Bruck loop, if α is a right [left] pseudoautomorphism with companion c , then c [ c 2 ] must lie in the left nucleus. In particular, for any such loop with trivial left nucleus, every right pseudoautomorphism is an automorphism and if the squaring map is a permutation, then every left pseudoautomorphism is an automorphism as well. We also show that every pseudoautomorphism of a commutative inverse property loop is an automorphism,...

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 ) ( 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 . 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 ) .