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We show that in a weak commutative inverse property loop, such as a Bruck loop, if is a right [left] pseudoautomorphism with companion , then [] must lie in the left nucleus. In particular, for any such loop with trivial left nucleus, every right pseudoautomorphism is an automorphism and if the squaring map is a permutation, then every left pseudoautomorphism is an automorphism as well. We also show that every pseudoautomorphism of a commutative inverse property loop is an automorphism, generalizing...
We investigate loops which can be written as the semidirect product of a loop and a group, and we provide a necessary and sufficient condition for such a loop to be Moufang. We also examine a class of loop extensions which arise as a result of a finite cyclic group acting as a group of semiautomorphisms on an inverse property loop. In particular, we consider closure properties of certain extensions similar to those as in [S. Gagola III, Cyclic extensions of Moufang loops induced by semiautomorphisms,...
Given a uniquely 2-divisible group , we study a commutative loop which arises as a result of a construction in “Engelsche elemente noetherscher gruppen” (1957) by R. Baer. We investigate some general properties and applications of “” and determine a necessary and sufficient condition on in order for to be Moufang. In “A class of loops categorically isomorphic to Bruck loops of odd order” (2014) by M. Greer, it is conjectured that is metabelian if and only if is an automorphic loop. We...
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