Transitive extensions of imprimitive groups

AbstractWe generalize the well-known Witt Theorem about the extensions of multiply transitive groups to imprimitive groups which are transitive on some t-tuples (t ⩾ 2) having a special property. A nontrivial application is also given

Composition of Permutation Representations of Triangle Groups

A triangle group is denoted by $\Delta(p,q,r)$ and has finite presentation $$\Delta(p,q,r)=\langle x,y | x^p=y^q=(xy)^r=1 \rangle .$$ We examine a method for composition of permutation representations of a triangle group $\Delta(p,q,r)$ that was used in Everitt's proof of Higman's 1968 conjecture that every Fuchsian group has amongst its homomorphic images all but finitely many alternating groups. We see that some of these compositions must give imprimitive representations, and in particular situations, where the representations being composed are all equivalent copies of an alternating group in the same degree, we can give a complete description of the structure of the composition of the representations. This article contains the main results of the author's PhD thesis