The Sylow p -subgroups of the symmetric group of degree p are just the cyclic subgroups generated by p -cycles. There are (p − 1)!/(p − 1) = (p − 2)! such subgroups simply by counting generators. The normalizer therefore has order p ⋅(p − 1) and is known as a Frobenius group F p (p −1) (especially for p = 5), and is the affine general linear group, AGL(1, p).