Structural interactions of conjugacy closed loops

• Published in: Transactions of the American Mathematical Society 2008-02, Vol.360 (2), p.671-689
• Main Author: DRAPAL, Ales
• Format: Article
• Language: English
• Subjects: Ales; Algebra; Automorphisms; Exact sciences and technology; Group theory; Group theory and generalizations; Homomorphisms; Integers; Isotopes; Mathematics; Research article; Sciences and techniques of general use; Uniqueness
• ISSN: 0002-9947; 1088-6850
• DOI: 10.1090/S0002-9947-07-04131-1

We study conjugacy closed loops by means of their multiplication groups. Let QQ be a conjugacy closed loop, NN its nucleus, AA the associator subloop, and L\mathcal L and R\mathcal R the left and right multiplication groups, respectively. Put M={a∈Q;M = \{a\in Q; La∈R}L_a \in \mathcal R\}. We prove that the cosets of AA agree with orbits of [L,R][\mathcal L, \mathcal R], that Q/M≅(Inn⁡Q)/L1Q/M \cong (\operatorname {Inn} Q)/\mathcal L_1 and that one can define an abelian group on Q/N×Mlt1Q/N \times \operatorname {Mlt}_1. We also explain why the study of finite conjugacy closed loops can be restricted to the case of N/AN/A nilpotent. Group [L,R][\mathcal {L},\mathcal {R}] is shown to be a subgroup of a power of AA (which is abelian), and we prove that Q/NQ/N can be embedded into Aut⁡([L,R])\operatorname {Aut}([\mathcal {L}, \mathcal {R}]). Finally, we describe all conjugacy closed loops of order pqpq.