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On the uniqueness of loops \(M(G,2)\). (English) Zbl 1101.20047

The paper is a contribution to Moufang loops of small orders. The author studies the construction of a Moufang loop \(M(G,2)\) from a group \(G\) given by O. Chein [Mem. Am. Math. Soc. 197 (1978; Zbl 0378.20053)] and shows that Chein’s construction is not at all arbitrary but unique in a way.
Given a group, eight multiplications (including the original one) can be created on the same carrier set by means of inverses and opposite operations. In the paper, the multiplications are described by permutations of \(G\times G\). Using transformation matrices of the multiplication table, a more general construction of a groupoid from a group is described. The resulting structures are quasigroups, and even might possess an identity element if the choice of permutations is suitably done. If this is the case the loop has the left inverse property, and if the loop satisfies the Bol condition it is Moufang.
For finite non-Abelian groups (different from elementary Abelian \(2\)-groups), four from the eight multiplications define groups while the remaining four yield proper (non-associative) Moufang loops, all (anti)isomorphic to \(M(G,2)\).

MSC:

20N05 Loops, quasigroups

Citations:

Zbl 0378.20053
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