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A loop X is said to satisfy Moufang's theorem if for every x , y , z ∈ X such that x (y z) = (x y) z the subloop generated by x, y, z is a group. We prove that the variety V of Steiner loops satisfying the identity (x z) (((x y) z) (y z)) = ((x z) ((x y) z)) (y z) is not contained in the variety of Moufang loops, yet every loop in V satisfies Moufang's theorem. This solves a problem posed by Andrew Rajah. [ABSTRACT FROM AUTHOR]

We introduce a class of non-Moufang loops satisfying Moufang's theorem. [ABSTRACT FROM AUTHOR]

A 6-cycle is said to be squashed if we identify a pair of opposite vertices and name one of them with the other (thereby turning the 6-cycle into a pair of triples with a common vertex). A 6-cycle can be squashed in six different ways. The spectrum for 6-cycle systems that can be squashed into Steiner triple systems has been determined by Lindner et al.(J Comb Des 22:189-195, ). The squashing of maximum packings of K with 6-cycles into maximum packings of K with triples has also been fully dealt with by Lindner et al. (Squashing maximum packings of 6-cycles into maximum packings of triples (submitted)). The object of this paper is to extend these results to minimum coverings of K with 6-cycles into minimum coverings of K with triples. We give a complete solution. [ABSTRACT FROM AUTHOR]