Displaying similar documents to “The free commutative automorphic 2 -generated loop of nilpotency class 3 ”

Commutators and associators in Catalan loops

Jan M. Raasch (2010)

Commentationes Mathematicae Universitatis Carolinae

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Various commutators and associators may be defined in one-sided loops. In this paper, we approximate and compare these objects in the left and right loop reducts of a Catalan loop. To within a certain order of approximation, they turn out to be quite symmetrical. Using the general analysis of commutators and associators, we investigate the structure of a specific Catalan loop which is non-commutative, but associative, that appears in the original number-theoretic application of Catalan...

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,...

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 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 → Inn Q determined by L x ↦ 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.