Some basic properties of planar Singer groups
A Singer group of a finite projective plane \(\pi\) is a group \(S\) of collineations of \(\pi\) which acts regularly on the set \(P\) of points of \(\pi\) and on the set \(L\) of lines of \(\pi\). By choosing a base point in \(P\), we identify the point set \(P\) with the group \(S\). A multiplier of \(S\) is an automorphism of the group \(S\) that is also a collineation of \(\pi\). Clearly, the multipliers of \(S\) form a group \(M(S)\). By construction, the action of \(S\) on \(\pi\) corresponds to the right regular action of \(S\) on itself. The author proves that the group \(S\) is abelian if and only if all left multiplications with elements of \(S\) are collineations of \(\pi\). After proving this result, the author comes to the main task of the paper under review; he studies the structure of the multiplier group \(M(S)\). The author proves the following results: 1) The group \(M(S)\) fixes a line of \(\pi\). 2) The 2-Sylow subgroup of \(M(S)\) is a cyclic direct factor of \(M(S)\) and, in particular, \(M(S)\) is solvable (The cyclicity of the 2-Sylow subgroups of \(M(S)\) was an earlier result of the same author). 3) There are some numeric bounds on the order of \(M(S)\) and on the order of an abelian subgroup of \(M(S)\).
scientific article; zbMATH DE number 806219
scientific article