Generalized quadrangles of order \((p,t)\) admitting a 2-transitive regulus, \(p\) a prime (Q855852)

Let \(S\) be a finite generalized quadrangle of order \((p,t)\), where \(p\) is a prime and \(t>1\). Assume that \(S\) admits a collineation group \(H\) which is doubly transitive on some regulus \(R\) and fixes each line of the opposite regulus. The author proves that \(S\) is isomorphic to one of the classical generalized quadrangles \(Q(4,p)\) or \(Q(5,p)\), or \(H\) induces a sharply 2-transitive group on \(R\) and \(p\) is a Mersenne prime.