On multiplication groups of loops

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The concept of multiplication groups of quasigroups was introduced by Albert Cl] and the connection between quasigroups and corresponding multiplication groups has been studied by Bruck [6], Smith [20] and Ihringer [14, 151. While studying the multiplication group of a loop Q (a quasigroup with neutral element) a central role is played by the stabilizer of the neutral element. This subgroup 1(Q) of the multiplication group is called the inner mapping group of Q. If Q is a group then it is clear that r(Q) consists of the inner automorphisms of Q. We also know that a loop Q is an abelian group if and only if 1(Q) = I. In this paper we study some properties of the inner mapping group and we also give a partial answer to the question: What are the multiplication groups of loops? This question is closely connected to certain transversal conditions. Sections 2 and 3 are devoted to investigating these conditions and in Section 4 we characterize multiplication groups of loops with the aid of these conditions. In the same section we prove one of our main results: If Q is a finite loop whose inner mapping group is cyclic, then Q is an abelian group. Finally, in Section 5 we use the properties of the inner mapping group in order to show that certain groups are not multiplication groups of loops. We also give examples of groups which are multiplication groups of loops. Our notation is standard and for basic facts about groups and loops WC refer to [4,7, 133. 112