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# HOMOGENEOUS SPACES

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# HOMOGENEOUS SPACES

DOI link for HOMOGENEOUS SPACES

HOMOGENEOUS SPACES book

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## ABSTRACT

A subquasigroup P of a quasigroup Q determines a homogeneous space P\Q. This space is defined as the set of orbits on Q of the relative left multiplication group of the subquasigroup P . If P is a subgroup of a group Q, then P\Q is just the set (2.10) of cosets of P in Q, and the group Q acts on P\Q by permutations specified by permutation matrices (Corollary 4.3). For a general quasigroup Q with subquasigroup P , there is an analogous action of elements of Q on P\Q by Markov matrices (4.14), the action being probabilistic rather than combinatorial. In mathematics, exact symmetry is understood conceptually as the action of

a group. For example, the symmetry of a square corresponds to the permutation action of the dihedral group Mlt(Z/4Z,−, 0) on the cosets of the subgroup Inn(Z/4Z,−, 0) — compare Example 2.2 and Section 2.4. Section 4.2 shows how the action of a quasigroup on one of its homogeneous spaces may be understood as an example of approximate symmetry, so-called macroscopic symmetry. The general version of this symmetry (for a finite quasigroup) is studied in Section 4.3. Section 4.4 considers regular homogeneous spaces, in which a quasigroup

acts on its own underlying set. The concluding Section 4.5 applies quasigroup homogeneous spaces in a new approach to issues concerning the breakdown of Lagrange’s Theorem for quasigroups.