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Paper IPM / M / 16306  


Abstract:  
Let A be an abelian group. If for any x,y 2A we define x . y Â¼2xy, then Ã°A,.Ã is a quandle called the Takasaki quandle and is denoted by T(A). In this article, we focus on the lattice of subquandles of a finite Takasaki quandle T(A), denoted by RÃ°AÃ. Indeed, we characterize the subquandles of T(A) and show that RÃ°AÃ is the lattice of cosets of A if and only if A is an odd abelian group or its 2Sylow subgroup is a cyclic group. Further, we determine the homotopy type of the lattice of subquandles of T(A) which is the wedge of r spheres of dimension d, where r is the number of prime numbers dividing jAj and d is determined by applying the fundamental theorem of finite abelian groups. Furthermore, we prove that RÃ°AÃ is graded if and only if A is an odd abelian group or its 2Sylow subgroup is cyclic or its 2Sylow subgroup is elementary abelian. Moreover, we show that the lattice of subquandles of a finite Takasaki quandle is determined by a unique Takasaki quandle. In other words, we show that the lattices of subquandles of two finite Takasaki quandles T(A) and T(B) are isomorphic if and only if A and B are isomorphic. However, the lattice of subquandles of a quandle which is not a Takasaki quandle can be isomorphic to the lattice of subquandles of a Takasaki quandle. We show this by providing a quandle Q of order 8 which is not a Takasaki quandle, but RÃ°QÃ is isomorphic to RÃ°Z8Ã:
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