Bol loops with non-normal nuclei. (Q1423806)

The purpose of this note is to present a large class of Bol loops with nonnormal nuclei. Such class is obtained by means of the following construction: given a group \(H\), a loop \(K\) and a homomorphism \(\Theta\colon K\to\Aut(H)\), then \(L:=H\times K\) becomes a loop together with the following binary operation: \[ (h_1,k_1)(h_2,k_2):=(h_1\Theta_{k_1}(h_2),k_1k_2), \] where \(\Theta_{k_1}\) is the automorphism of \(\Aut(H)\) associated to \(k_1\) by the automorphism \(\Theta\) (notice that such a \(\Theta\) always exists, for instance the homomorphism sending every element of \(K\) into the identity automorphism of \(H\)). We set \(L=:H\times_\Theta K\). By previous results it is known that if \(K\) is a Bol loop which is not Moufang, then \(L\) is again a Bol loop which is not Moufang. Moreover, the author proves that \(N(L)=H\times N(K)\vartriangleleft L\) if and only if \(N(K)\vartriangleleft K\) (where \(N(L)\) (resp. \(N(K)\)) denotes the nucleus of \(L\) (resp. \(K\)). Hence if \(K\) is a Bol loop with a nonnormal nucleus then \(H\times_\Theta K\) has the same property as well. Thus, starting for instance from the examples of Bol loops with nonnormal nuclei \(K\) presented by \textit{D. A. Robinson} and \textit{K. H. Robinson} [Arch. Math. 61, No. 6, 596-600 (1993; Zbl 0815.20069)], the present construction provides many other examples for any choice of the group \(H\) and the homomorphism \(\Theta\). The final corollaries of the paper yield further generalizations of the basic construction.