A note on a lemma of Brauer (Q1842159)

Let \({\mathcal S} = ({\mathcal P}, {\mathcal B}, {\mathcal I})\) be an incidence structure. The structure \(\mathcal S\) is called symmetric if \(| {\mathcal P} | = | {\mathcal B}|\). The rank of a structure \(\mathcal S\) is equal to the linear rank of its incidence matrix over the rationals. It is called a structure of maximal rank if its rank is equal to the number \(| {\mathcal P}|\). \textit{R. Brauer} [Ann. Math., II. Ser. 42, 926-935 (1941; Zbl 0061.037)] proved that permutations induced by an automorphism of the maximal rank symmetric structure \(\mathcal S\) on \(\mathcal P\) and \(\mathcal B\) are similar. The authors of this note generalize a result of Brauer for symmetric structures.