We start from the embedding of the Klein model of a hyperbolic plane H over a Euclidean field K in its direct motion group G := PSL_2 (K) and of both in PG(3, K). We present a geometric procedure to obtain loops which are related to suitable regular subsets of direct motions as transversals of the coset space G/D, where D is the subgroup of hyperbolic rotations fixing a given point o ∈ H. We investigate some properties of such loops and we determine their automorphism groups.
A Geometric Environment for Building up Loops
PASOTTI, Stefano;ZIZIOLI, Elena
2015-01-01
Abstract
We start from the embedding of the Klein model of a hyperbolic plane H over a Euclidean field K in its direct motion group G := PSL_2 (K) and of both in PG(3, K). We present a geometric procedure to obtain loops which are related to suitable regular subsets of direct motions as transversals of the coset space G/D, where D is the subgroup of hyperbolic rotations fixing a given point o ∈ H. We investigate some properties of such loops and we determine their automorphism groups.File in questo prodotto:
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loop-sezione-res-math.pdf
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