The structure of free automorphic Moufang loops. (Q2845036)

An automorphic Moufang loop \((M,\cdot)\) is a Moufang loop which is also an A-loop, namely its inner mapping group consists of automorphisms of \((M,\cdot)\). In this paper some relevant properties of such a loop are studied.NEWLINENEWLINE In Section 2 the authors investigate the subgroup \(A(M)\) of the associators, the subgroup \([M,M]\) of the commutators and the subgroup \(D(M)=A(M)\cap[M,M]\). In particular they prove that if \(D(M)=1\) then \(M\) is a subdirect product of a commutative Moufang loop with a group.NEWLINENEWLINE In Section 3 they consider the following construction: let \(\mathcal F_n\) be the free group generated by \(X=\{x_1,x_2,\ldots,x_n\}\), let \(\mathcal C_n\) be the free commutative Moufang loop generated by \(Y=\{y_1,y_2,\ldots,y_n\}\) and let \(z_i=(x_i,y_i)\in\mathcal F_n\times\mathcal C_n\) for \(1\leq i\leq n\). Let \(\mathcal A_n\) be the subloop of \(\mathcal F_n\times\mathcal C_n\) generated by \(Z=\{z_1,z_2,\ldots,z_n\}\). Then the following holds true:NEWLINENEWLINE (i) \(\mathcal A_n\) is a loop with \(D(\mathcal A_n)=1\),NEWLINENEWLINE (ii) If \(\mathcal M_n\) is a free automorphic Moufang loop of rank \(n\) then \(\mathcal M_n\) is isomorphic to \(\mathcal A_n\).NEWLINENEWLINE The latter statement is the main result of the paper since it allows a complete description of the structure of this class of loops.