Let \(G\) be a finite group and \(H\) be a subgroup of \(G\). A normalized right transversal of \(H\) in \(G\) is a subset \(S\) of \(G\) obtained by selecting one and only one element from each right coset of \(H\) in \(G\) and such that \(1 \in S\). If \(S\) is a normalized right transversal of \(H\) in \(G\), then \(S\) can be equipped with a binary operation \(\circ\) by defining \(\{x \circ y\}=S \cap Hxy\) for \(x,y \in S\). The structure \((S, \circ)\) is a right loop with identity \(1\), that is, a right-quasigroup with both-sided identity.
Two normalized right transversal \(S\) and \(T\) of \(H\) in \(G\) are isotopic if their induced right loop structures are isotopic. Let \(\mathsf{Itp}(G, H)\) denote the set of isotopic classes of normalized right transversal of \(H\) in \(G\). It has been proved by V. Kakkar and the second author [Algebra Colloq. 23, No. 3, 409–422 (2016; Zbl 1366.20017)] that if \(G\) is a finite nilpotent group and \(H \leq G\) is such that \(|\mathsf{Itp}(G,H)|=1\), then \(H\) is normal in \(G\).
The main theorem of the paper under review asserts that the previous result continues to hold if \(G\) is assumed to be solvable.