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On left conjugacy closed loops with a nucleus of index two

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Abstract

A loop Q is said to be left conjugacy closed (LCC) if the left translations form a set of permutations that is closed under conjugation. Loops in which the left and middle nuclei coincide and are of index 2 are necesarilly LCC, and they are constructed in the paper explicitly. LCC loops Q with the right nucleus G of index 2 offer a larger diversity, but that is associated with the level of commutativity of G (amongst others, the centre of G has to be nontrivial). On the other hand, for each m ≥ 2 one can construct an LCC loop Q of order 2m in such a way that its left nucleus is trivial, and the right nucleus if of order m. If Q is involutorial, then it is a Bol loop.

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Author information

Correspondence to A. Dräpal.

Additional information

Work supported by institutional grant MSM 113200007 and by Grant Agency of Czech Republic, Grant 201/02/0594. The paper was written while the author was visiting Universitaet Hamburg in January 2004.

A. Kreuzer

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Dräpal, A. On left conjugacy closed loops with a nucleus of index two. Abh.Math.Semin.Univ.Hambg. 74, 205 (2004). https://doi.org/10.1007/BF02941536

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2000 Mathematics Subject Classification

  • Primary 20N05
  • Secondary 08A05

Key words and phrases

  • Left conjugacy closed loop
  • nucleus