On groups with free transitive action on the set of flags of a projective plane (Q1357993)

Let \((P,L,I)\) be a projective plane. A group \(\mathcal G\) acts on the set \(I\) of flags freely (transitively) provided that \(g(p,l)=(p,l)\) implies \(g=1\) (if for all \((p_1,l_1)\in I\) and \((p_2,l_2)\in I\), there is \(g\in{\mathcal G}\) such that \(g(p_1,l_1)=(p_2,l_2)\)) with \(p,p_1,p_2\in P\), \(l,l_1,l_2\in L\). Some criteria for transitive free actions of a group \(\mathcal G\) on the set of flags of a finite Desarguesian plane were given in the paper by \textit{D. G. Higman} and \textit{J. E. McLaughlin} [Geometric ABA-groups, Ill. J. Math. 5, 382-397 (1961; Zbl 0104.14702)] in terms of subgroups of \(\mathcal G\). In this case \(\mathcal G\) has order \(3\times 7\) or \(3^2\times 71\) and such groups are not simple. In the paper under review the existence of an infinite simple group acting freely and transitively on the set of flags of some projective plane is proved. Such a group satisfies the following conditions: (i) if \(\mathcal A\) is the stabilizer of a point and \(\mathcal B\) is the stabilizer of a straight line then \(\mathcal A\) and \(\mathcal B\) are free groups of rank \(2\); (ii) the groups \(\mathcal A\) and \(\mathcal B\) are maximal in \(\mathcal G\) and there exists an \(m\) such that \({\mathcal G}=(g{\mathcal A}^m)\) (\({\mathcal G}=(g{\mathcal B}^m))\) for every \(g\in{\mathcal G}\setminus{\mathcal A}\) \((g\in{\mathcal G}\setminus{\mathcal B}\)); and (iii) there exists an \(m\) such that \({\mathcal G}={\mathcal C}^m\) for every nonidentity class \(\mathcal C\) of conjugate elements. The construction of the group is based on corepresentations of \(\mathcal G\) with four generators and on some infinite set of defining relations of the form \(R_i=1\), \(i\in{\mathcal N}\).