C-loops: extensions and constructions

Details C-loops: extensions and constructions C-loops: extensions and constructions

2004-12-19

arxiv.org/abs/math/0412390v1

J. Algebra and its Applications 6 (2007), 1-20

Mathematics

Group Theory

17 pages, amsart

Scientific paper

C-loops are loops satisfying the identity $x(y\cdot yz) = (xy\cdot y)z$. We develop the theory of extensions of C-loops, and characterize all nuclear extensions provided the nucleus is an abelian group. C-loops with central squares have very transparent extensions; they can be built from small blocks arising from the underlying Steiner triple system. Using these extensions, we decide for which abelian groups $K$ and Steiner loops $Q$ there is a nonflexible C-loop $C$ with center $K$ such that $C/K$ is isomorphic to $Q$. We discuss possible orders of associators in C-loops. Finally, we show that the loops of signed basis elements in the standard real Cayley-Dickson algebras are C-loops.

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