Solvable groups and loops (Q1911789)

Let \(Q\) be a loop and \(M(Q)\) be its multiplication group, i.e. the subgroup of the symmetric group \(S_Q\) generated by all the left and right translations. And moreover, let us assume that there exists a series of subloops of the loop \(Q\) of the form \(1=Q_0\leq Q_1\leq\cdots\leq Q_n=Q\), where \(Q_i\) is a normal subloop in \(Q_{i-1}\) and \(Q_i/Q_{i-1}\) is an Abelian group for all \(i=1,\dots,n\). Then \(Q\) is called solvable loop. It is easy to construct solvable loops with non-solvable multiplication groups -- the smallest example is of order 10. In the paper under review, the following main result is proved as Theorem 1: If \(Q\) is a finite loop such that \(M(Q)\) is solvable, then \(Q\) is solvable. The paper is concluded with some applications and examples.