On finite loops with nilpotent inner mapping groups. (Q604211)

Groups which are isomorphic to inner automorphism groups of groups are called capable groups. The question which Abelian groups are capable was solved by \textit{R. Baer} [Math. Z. 38, 375-416 (1934; Zbl 0009.01101)]. In the finite case if \(H=C_1\times C_2\times\cdots\times C_n\) is a finite Abelian group written as a product of non-trivial cyclic groups \(C_1,C_2,\dots,C_n\) such that \(|C_{i+1}|\) divides \(|C_i|\) for each \(i\), then \(H\) is a capable group if and only if \(n\neq 2\) and \(|C_1|=|C_2|\). The authors refer to this result as the Baer condition. In this paper there is the question: Is it possible to extend the Baer condition to be true for Abelian inner mapping groups of finite loops? The authors show that if \(Q\) is a finite Moufang loop of odd order, then the Baer condition is satisfied for an Abelian group \(I(Q)\).