Frobenius groups with many involutions (Q1849895)

It is well known that the sharply 2-transitive groups are an important class of Frobenius groups and they can be represented algebraically by means of the so-called neardomains. In this paper the author shows that a Frobenius group can be obtained starting from a left loop \((L,\cdot)\) and a particular subgroup \(T\) of \(\text{Sym }L\), called transassociant of \(L\), acting fixed-point-freely on \(L\setminus\{1\}\). Indeed, he proves that the quasi-direct product \(L\rtimes_QT\) is a Frobenius group and conversely that every Frobenius group \((G,\cdot)\) acting on a set \(P\) is isomorphic to \(L\rtimes_Q\Omega\) where \(\Omega\) is the stabilizer of a fixed element \(e\in P\) and \(L\) is a set of transversals (i.e. a set of representatives of the left cosets of \(\Omega\) in \(G\) fulfilling a suitable property) which always carries a left loop structure. Such correspondence is particularly interesting in the case of a Frobenius group with many involutions (see the definition in the note): in this case the associated left loop \(L\) becomes a \(K\)-loop. Specifically, if \(G\) is a sharply 2-transitive group then \(G\) falls under this class of Frobenius groups and the associated \(K\)-loop is isomorphic to the additive loop of the corresponding neardomain. The aim of the author is to set forth a new point of view in the direction of finding examples concerning the still problem of the existence of a proper neardomain.